Songbooks of the 18th and early 19th century that I've seen,
primarily in the Library of Congress, and Folger Shakespeare
Library. For a much more complete listing see old CBEL section on
Poetical Miscellanies. There are a few here that are not in the
CBEL listing.

Sources: LCMD-Library of Congress, Music Division; LCRB-Library
of Congress, Rare Book; LCM-Library of Congress, Main; Folger-
Folger Shakespeare Library.

Seen, With Music: 

17th cent.
A choice collection of 180 loyal songs, London, 1685. LCMD.

18th cent.
Orpheus Caledonius or a Collection of the best Scotch Songs set 
  to Musick by W. Thomson. London. n.d. [1725/6] [Copy seen is at
  Univ. of Brit. Columbia, Canada] See 1733. 
Songs in Flora's Opera, (1730), (Folger) This deletes 5 tunes 
  from Flora, 1729. [pirated as, Songs in Flora, 1737. (Lib. of 
  Congress)] Tunes are better arranged to fit songs than in the 
  ballad opera.
The Merry Musician; Or, A Cure for the Spleen, 2vols, 2nd edit.  
  London: Walsh and Hare, Vol. I, 1730, II, n.d. w/music LCMD   
  [Bound with Songs in The Lottery]
Songs in The Lottery, #300, np, nd (Catalog # is that of J.   
  Walsh, and is c 1732. LCMD. [bound with Vol. II of the Merry   
  Musician, 1730. Same songs as in the ballad opera. Some tunes 
  better arranged.]
Merry Mountebank; or, The Humorous Quack-Doctor. By Timothy 
  Tulip. London: W. Pearson. 1732. w/music. LCMD. [BL copy is not 
Orpheus Caledonius. By William Thomson. 2 vols. 1733. Facsimile 
  print, 1962. [1st vol. has same tunes but not all are same 
  songs as 1725/6 ed.] 
The British Musical Miscellany, 6 vols. London: J. Walsh, n.d. [1
  & 2, 1734; 3 & 4, 1735; 5 & 6, 1736] LCMD [Among the songs are
  reprints most of the peices in 1733 ed. of Orpheus Caledonius] 
Universal Musician, London: for the Booksellers, c 1737. w/music.
  engravings with illustrations, mostly by J. Smith and H. 
  Roberts. LCMD. 
The Merry Medley, or a Christmas Box for Gay Gallants and Good  
  Companions, 1744, II -1745. [A few songs and about a dozen   
  country dances with tunes in each] Univ. of Wisconsin, seen, 
  and Huntington Library, not seen.  
Calliope, or English Harmony, London: Henry Roberts, c 1746. 
  w/music. Folger. [earlier ed. 1739] [Looks like collection of 
  ss songs with engraved illustrations, many sheets dated 1739, 
  from earlier ed.] 
Universal Harmony, London: J. Newberry, 1746. w/music. Folger
Spinnet; or Musical Miscellany, London: J. Wren, 1750. w/music. 
The Muses Delight, Liverpool: John Sadler, 1754. w/music. Folger,
  LCMD [This is quite common]
Apollo's Cabinet, Liverpool: J. Sadler, 1757. [Muses Delight, 
  1754, with additions] LCMD
Thirty Scots Songs Adapted for a Voice and Harpsichord, The Words
   from Allan Ramsay. Edinburgh, By Robert Bremner. n.d.: A 
   Second Set of Scots Songs. Words from Allan Ramsay. By Robert
   Bremner. Edinburgh, nd. (c 1757) LCMD.
Orpheus Britanicus, or, The Gentleman's and Lady's Musical 
  Museum. London: Benj. Cole. 1760. w/music. LCMD
A Collection of the Best old Scotch and English Songs. By James 
  Oswald. n.d. [1761?] LCMD.   
Clio and Euterpe, 3 Vols. London: Henry Roberts. 1762. w/music. 
Select Collection of Vocal Music, London: J. Johnston, c 1770. 
  New songs with music by Italian composers. LCMD
Vocal Music or, the Songster's Companion, 4 vols. London, 1st vol
  only, LCMD, R. Horsfield, 1772-75 LC [but LCMD couldn't locate
  a catalogued 4 vol set.]
A Collection of Scots Songs adapted for a Voice and Harpsichord.
   By Neil Stewart. Three issues in Library of Congress all have 
   different addresses in Edinburgh. Printed and Sold by Neil 
   Stewart at his Shop: (1) Parliament Square, (2) Millno Square
   Opposite to the Tron Church, (3) No. 37 South Bridge St. pp. 
   28. The one apparently oldest is (1), where an anonymous MS 
   note states it is original ed. of 1772. 
Vocal Music, or The Songster's Companion, London: J. Bew, c 1778,
  w/music. Folger, LCMD 
Thirty Scots Songs. Adapted for a Voice and Harpsicord. The Words
   by Allan Ramsay. Edinburgh. By Neil Stewart. n.d. (I-III) 
   [1781] First 2 in LCMD, and all in Newberry Library, Chicago.
   [But I did not look at the latter, and don't know whats in 
   Vol. III]
The Convivial Songster, London: John Fielding, [1782]. w/music. 
  LCMD, U. Wisc. 
The Vocal Enchantress, London: J. Fielding, 1783. w/music. LCMD.
Musical Miscellany, Perth: J. Brown, 1786. w/music. [Edinburgh 
  issue without printer's name, same date, both LCMD.]
The Scots Musical Museum, 6 vols. Edinburgh; J. Johnson, 
  [1787-1803/4]. w/music. I, 1787; II, 1788; III, 1790; IV, 1792;
  V, 1797; VI, 1803 [actually 1804] LCMD 
Edinburgh Musical Miscellany. [I], 1792, II, 1793 Edinburgh. 
  LCMD. w/music.
A new and complete collection of the most favourite Scots songs.
  D. Corri. 2 vols. Edinburgh: Corri and Sutherland, [1788]; 
  Corri and Sutherland, [c 1790]; Corri and Coy, [c 1793]. 
  w/music. LCMD has late issue, and some other songbooks issued 
  by Corri. 
Calliope, or The Musical Miscellany, London: C. Elliot and T. Kay
  and Edinburgh: C. Elliot. 1788, Folger, LCMD. w/music. [common]
Edwin's Pills to Purge Melancholy (all the songs sung by Mr. 
  Edwin). London, 1788. w/music. Folger.
A Collection of Favorite Songs; A. Reinagle, Philadelphia, c 
  1789. [mostly from operas by Arnold, Shield, etc.] w/music. 
A Selection of Scots Songs, P. Urbani. 4 vols. Edinburgh: For 
  the Author. 1792-c 1794. [1st 3, later issue, in Lib. of 
  Congress] w/music. Edinburgh: Urbani and Liston. 
Vocal Magazine, 1-3, Edinburgh: C. Stewart and Co. 1797, -98, 
  -99. w/music. LCMD.
Musical Repository. Glasgow: Adam and Carrick, 1799. w/music. 

19th cent.
Yorkshire Musical Miscellany, Halifax: C. Jacobs, 1800. w/music.
The British Musical Miscellany. Edinburgh: Thomas Turnbull, 1805,
  w/music. LCMD 
Songster's Favourite Companion. London for Lane & Newman and for
  A. Macgoun, Glasgow, n.d. w/music [owner entered his name and 
  date of 1803] LCMD
British Melodies, Polyhymnian Co. [London, George Walker] n.d. [c
  1810] Engraved like single sheet songs. LCMD 
Cabinet of Harmony, London: Printed for Oliver & Boyd, Edinburgh.
  1814. LCMD 
Irish Minstrelsy, J. Smith. London: Goulding and Co. [1814] 12 
  Irish tunes with new verses by Ed. Fitzsimons. LCMD
British Orpheus, The, London, c 1817. w/music. LCMD
The Scottish Minstrel, R. A. Smith. 6 vols. Edinburgh: Robt. 
  Purdie 1820-25?. LCMD
Selection from the Ancient Music of Ireland, A. Chiefly composed
  by John M'Creery. Petersburg: Yancy and Burton, 1824.
  w/music. LCMD [Title misleading, new song to older Irish tunes 
Miniature Museum of Scotch Songs. 2 Vols. Edinburgh: Walter & 
  Anderson, 1816-25. w/music LCMD
The Melodist, .. English Scotch, and Irish songs. New York: W. &
  P. C. Smith. 1824. LCMD
The Irish Minstrel, R. A. Smith. Edinburgh: Robt. Purdie. 1825 
  w/music. LCMD; [This edition supposedly suppressed for 
  copyright violations. Songs with tunes from Stevenson/ 
  Moore's A Selection of Irish Melodies were replaced in next 
  edition. Whether this latter was styled 1st ed. or 2nd I do not
  know. I have seen only an undated 2nd. edition at Catholic 
  Univ. of America. Elsewhere this work (as 1st ed.?) is said to
  be of 1828.  
Linnet, London, T. Tegg, 1831 LCMD?
Thrush, The, London: Thomas Tegg. 1827. w/music. Folger
London Minstrel, London, Dean and Munday, 1825. w/music. LCMD 
Musical Casket, [Running title, tp missing, c 1840? LCMD [Gore: 
  James Macewan, A Musical Casket of Melodies for the Millions, c

Seen, without Music:

17th cent. 
The New Accademy of Compliments, London: S. Rand, 1669. Folger 

18th cent.
Mirth diverts all care. 1708. wo/music. Folger.
The Hive, Vol. I & II, 1724; Vol. III, 1725; Vol. IV, 1732 (LCRB)
  The Hive, 4th edition, Vol. I, 1732; Vol. II, 1733; Vol. III,
  n.d. (c 1733-4) (LCRB). Vol. III, 1725 also at Univ. of Wisc. 
The Choice, Vol. III, London  [1733?] LCMD.
Vocal Miscellany, 2nd ed., 2 vols. London, Joseph Hazard, 1734. 
  wo/music LCMD [LCMD has Vol. II cataloged separately, latter 
  not listed in old CBEL.] 
Pinkethman's Jests, Part II, Songs. London: John Atkinson, 1735 
  [wo/music. Most songs from Pills] LCRB
Nightingale, London: J. Osborn. 1738. wo/music. [good coll'n of 
  422 songs] Folger. 
The Musical Companion, or Lady's Magazine, 1741. wo/music. LCMD,
Ladies Delight, or The Merry Songster. London: T. Wright, 1741. 
  [tunes named but no songs are familiar. All new by one hand?] 
Philomel; being a Small Collection of Only the Best English 
  Songs, London: M. Cooper, 1744. wo/music. LCMD [BL, and Nat'l 
  Union Catalog lists copy at Yale. This seems to be the fisrt 
  book to print "John Anderson, my Jo", but far from the last] 
The Canary Bird, London: Printed in the Year 1745. wo/music. 
Warbling Muses, by Benjamin Wakefield. London: G. Woodfall. 1749.
  wo/music. [731 songs of which about 3 are familiar. Did 
  Wakefield write the rest?] Folger.
A Collection of Loyal Songs, Poems, 1750. no imprint. Folger. 
The Merry Companion, or Universal Songster, 4th ed. with 
  additions. London: W. Johnston, 1750. wo/music. Folger.
Wreath, London: W. Reeve, A. Dodd, & E. Cooke. 1752. wo/music. 
Muses Banquet, Reading: C. Mickleworth, et al., 1752. wo/music. 
The Scoundrel's Dictionary (with several cant songs at end), 
  1754. wo/music. Folger, LC.
The Comic Miscellany, 2 vols. [V. 1 only is songs, expanded 
  version of Philomel, 1744] London: M. Cooper, 1756. Folger 
Vocal Medley, or Universal Songster, York: John Jackson, 1756 
  [wo/music, almost all theater songs] LCMD
Orpheus, songs; Vol. I, The Linnet; Vol. II, The Thrush; Vol. III
  The Robin, all 1749. [wo/music or tune indication] LCRB
The Merry Fellow, or Jovial Companion, By Luke Lively, Gent. 
  Dublin: James Hoey. 1757. wo/music. [Miscellany, with fifty 
  songs, a few of which are Irish, and many bawdy riddles] 
The Musical Companion, London: Stevens and Seymour, 1759. Folger.
Syren, London: S. Blandon & F. Blyth c 1760. wo/music. Folger.
The Polite Songster, London: J. Ross. wo/music. 1760. LCMD.
The Charmer, 3rd ed. [contents of a second volume listed at 
  beginning, but was it ever published?] wo/music. Edinburgh: M.
  Yair, 1765. LCRB
The Brent; or, English Syren, London: S. Bladon, 1765. LCMD 
  wo/music or tune direction, mostly theater songs.
The Muses Delight, or, The London Polite Songster. London: J. 
  Pridden, 1766. U. Wisc.
London Songster, The, or Polite Musical Companion, London: W. 
  Nicoll, 1767. wo/music. Folger.
The Masque, London: Richardson and Urquhart. nd. 331 songs. 
  [Folger puts it at 1767, so this is possibly the 1st editon or
  a reissue of it, but spelling of some words are as in c 1770, 
  and c 1777 editions, which differ from those in 2nd. ed of 
  1768. Adv. in Boston, 1767, but no American copy is known for 
  sure. Another source put 1st ed at 1765, but I don't know for 
  sure if 1st edition was actually dated. See 1768, 1770, 1777] 
The Masque, 2nd. ed. London: Richardson and Urquhart. 1768. 
  Folger. wo/ music. 337 songs. See 1767, 1770, 1777. 
Polyhymnia, London: F. Newberry. 1769. wo/music. Folger. [mostly
  theater songs]
The Masque, nd. London: Richardson and Urquhart. [CBEL, 1770?]
  342 songs. Folger [where it is put at c 1769. See 1767, 1768, 
Jovial Companion, or, The New Sentimental Songster. London, M. 
  Cooper, n.d. wo/music. c 1770, Folger.
The Humours of London, London: T. Cooke, c 1770. wo/music. LCRB 
  [Several songs by G. A. Steven, apparently earlier than in his
  collecton of 1772, below. A rousing bacchanalian is "The True 
  Englishman" to the tune of "Shawnbree" (Sean Buidhe/ Over the 
  water to Charlie), which seems to imitate the song "Larry 
  Grogan" or "The Rakes of Mallow"] 
Vocal Companion, London sold by the Booksellers. c 1770. 
  wo/music. Folger.
Bacchus and Venus; Or, The Harmony of Love and Wine, London, n.d.
  [CBEL c 1770] wo/music. Folger.
A New Academy of Compliments, Glasgow, 1772. Folger. wo/music. 
  [This is a curious reprint of some edition of the 1720's. No 
   songs are later.]
The Warbler, London; or, Sweets of Harmony selected. London: 
  Booksellers, 1772. LCMD. 
Songs, Comic, and Satrical. By George Alexander Stevens. London:
  For the Author. 1772. wo/music, with tune directions. LCMD. 
  [There is a later edition, not seen]
The New Merry Companion, or Complete Modern Songster, London: 
  John Wheble [1772?]
The Ladies Polite Songster; or Harmony of the Fair-Sex, London: 
  T. Shepherd and Stephen Smith, n.d. [1772] wo/music. Folger.
Bull-finch, London: Baldwin, Horsfield and Wilkie. [c 1775] LCMD. 
Bucks Bottle Companion, The, London, 1775. Folger, LCMD. 
Charms of Melody, Dublin, John Colles, 1776. Folger. wo/music
The Hummingbird, London, J. Johnson, 1776. Folger, LCMD. 
The Masque, nd. London: Richardson and Urquhart. [CBEL, 1777?] 
  354 songs. Folger [see 1767, 1768, 1770. Folger puts this at c 
The Linnets, The, Or, Songster's Companion, Wolverhampton, 1777,
  and 1780. Folger. wo/music.
Vocal Magazine, or British Songster's Miscellany, Vol. the first.
  J. Harrison, London, 1778. = Vocal Magazine, Or, Compleat
  British Songster, London, Harrison & Co. 1781. Folger. LCMD. 
  1286 songs, no titles or tune indication. Also 1784, LCMD.
The Scots Nightingale, 2nd. ed. Edinburgh: James Murray. 1779. 
  wo/music. LCMD [1st ed. of 1778 not found]
St. Cecelia, Edinburgh: C. Wilson, 1779. [wo/music, and few tune
  directions]. LCRB
The Goldfinch, Glasgow, J. & M. Robertson, 1780. Folger, LCMD. 
Scots Vocal Miscellany, Edinburgh: J. Roch, 1780. wo/music LCRB
Charms of Chearfulness, or Merry Songster's Companion, London: W.
  Lane, 1781. Folger. wo/music. [Contains the song "Larry Grogan"
  which the Irish seem to have lost. See next.]
The Polite Singer: a Collection of Choice Song, North-Shields: 
  Mathew Brown, 1781. LCMD wo/music. [Contains "Larry Grogan", 
  see previous.]
The Modern Syren, or Enchanting Songstress, Sunderland, J. 
  Graham. 1781. Folger. wo/music.
The Charmer, 2 Vols. 1st is 4th ed. 2nd says entire new 
  collection. Edinburgh: J. Sibbald, 1782. LCMD
Saint Cecelia, or British Songster, Edinburgh, 1782. wo/music 
  LCMD. [see 1779]
Modern Songs. By John Free, Birmingham: for the Author, 1780. 
  wo/music [Folger?]
Roundelay, or the New Syren, A new edition. London: Wm. Lane. c 
  1783. Folger. wo/music 
The British Muse, by John Vint, 2nd ed., T. Saint, Newcastle, 
  1784. Folger. wo/music
The London Songster, or Polite Musical Companion. London, T. 
  Lowndes and Son, a new edit. 1784. wo/music. LCMD.
The Minstrell, London, c 1785. Richardson and Urquhart. Folger, 
  LC. wo/music.
New Vocal Miscellany (songs by the author, generally pretty good,
  several to Irish tunes), London, By Wm. Collins, 1787. Folger.
Songster's Companion, The Second Edition. Coventry: M. Luckman. c
  1787. [wo/music. mostly theater songs.] LCRB
Parsley's Lyric Repository for 1788. Folger. wo/music. LCMD
New Vocal Enchantress, London, 1789. Folger. wo/music
Charms of Chearfullness; or Merry Songster's Companion, 4th ed. 
  London: W. Lane. 1789. wo/music. LCMD M1619.C5
The Billington, or Town and Country Songster London: E Wenman. 
  wo/music (mostly theater songs) 1790. LCMD.
The Busy Bee, or, Vocal Repository, c 1790. Vol. III only. 
  Section of Scotch and Irish Songs. wo/music LCMD.
Festival of Anacreon, 7th ed. London, George Peacock, 1789, 
  Folger: nd / [c  1791] LCMD. wo/music. [fake edition # and 
  publisher, 1st here is prob. 2nd edit.]
Merry Companion, or Feast for the Sons of Comus, London: W. Lane,
  c 1791. LCRB [CBEL notes 1786 ed., but this edition contains 
  songs of 1790.]
Parsley's Lyric Repository for 1791. [London]: R. Parsley. [1791]
  wo/music LCMD  M1619.P26 
Paddy Whack's Bottle Companion, 1791. London: Wm. Holland. 
  [Closely related to Festival of Anacreaon, with same typography
  and many of the same pieces. But "Paddy Whack" is not given.] 
The Banquet of Thalia, York: Atkinson, 1792 Folger. w/o music
The General songster, London, 1793. Folger. wo/music.
Songster's Repository; or, Convivial Companion For the Year 1793.
  London: for the Author and Compiler. 1793 [almost all are 
  recent theatre songs. 4 Irish. wo/music] LCRB
Festival of Momus, a Collection of Comic Songs, London, W. Lane.
  nd [1793? earlier edit. of this is of c 1793.] wo/music. LCMD.
Sweet Robin, or, The Children in the Wood, London, 1794. 
  wo/music. Folger.
Whim of the Day for 1794, J. Roach, London, LCMD
Whim of the Day for 1795. J. Roach, London, LCMD
The British Songster; or Pocket Companion. London: William Lane.
  n.d. [c 1794] LCMD. wo/music, mostly theater songs.  
London Songster, The, or Musical Boquet, London: Wm. Lane. c 
  1795. Folger. wo/music.
The Polite Songster, or, Vocal Melody, London, Wm. Lane. c 1795.
  Folger. wo/music. LCMD.
Charms of Melody, or Siren Medley, Dublin. 1795-1810, 100 four 
  page issues, (3 missing) issued approximately periodically. 
  Folger. wo/ music. [This bound set formerly that of J. W. 
  Ebsworth] An issue of c 1806-7 contains a text of "The Girl I 
  left behind me" with no mention of a 'Brighton Camp' and this 
  is the earliest text of the song, or songs, that I've seen.
The Universal Songster, London, c 1796-98. Folger. wo/music. [No
  relation  to the 3 vol. collection with same title, 1825-28, 
  and its reissues]
Hooper's Whim for 1799, London, 1799. wo/music. LCMD
Whimsical Songster, Manchester: Hopper & Co. n.d. [c 1799]. LCMD 
Whim of the Day for 1799, The, London: J. Roach. wo/music. LCMD.

19th cent.
New Musical Charmer, Glasgow: James Mundel. 1800 LCMD [no music 
  or tune directions, Songs from Scots Musical Museum, O'Keeffe's
  plays and operas and a few others, e.g. "Barbadoes Bell(e)s"]
Songster's Museum. Gosport: W. M. Martin. 180-, LCMD  
The Convivial Repository. London: J. Painter, 1801. LCMD?
Apollo's Budget. London: Crosby and Letterman, 1801. wo/music 
New Songs. The Fashionable Songster, Manchester: Hopper and Hope,
  1801, LCMD 
Banquet of Bacchus, London: Crosby and Letterman. 1802. LCMD 
  [nothing of interest, to me at any rate]
Whim of the day (for 1802, -03, -04, and -05). U. Wisc. 
The Little Warbler, 2 vols, London, 1804. LCMD. wo/music
The Musical Banquet, The second edition of, Glasgow, A Maccoun c
  1801-3 [original 1800, CBEL  LCMD]
Rural Songster, Edinburgh: Oliver & C0., 1806. wo/music LCM
Oliver's New Selection of Comic Songs, Edinburgh: Oliver & Co., 
  1806. LCMD. wo/music.
Oliver's Choice Selection of Comic Songs, Second Edition, 
  Edinburg: Oliver & Co. 1807. LCMD. wo/music.
The Shamrock, or Hibernian Songster, Edinbugh: Oliver & Co. nd 
  [1803-8?]. LCMD. wo/music.
Apollo, The. Ipswich, c 1810. wo/music. U. Wisc.
John Bull; or the True Briton, Penrith, c 1810. wo/music. Folger
Mirth & Melody, Northampton: J. Freeman, c 1810. wo/music. LCMD.
Northumbrian Minstrel. Alnwick: W. Davidson, 1811. wo/music. LCMD
  (Many by R. Burns)
Banquet of Thalia, York: Thos. Wilson & Son. 1812. LCMD 
The Union Imperial Songbook, Edinburg: A. Hogg, John Robertson, 
  Macreidie & Co., and George Cowies & Co., nd. London. LCMD 
The Harp of Erin, Dublin, 1816. LCMD. wo/music
The Pocket Encyclopedia of Scottish, English and Irish Songs. 2 
  vols. Glasgow: Andrew and James Duncan. 1816. wo/music [Nine 
  songs by Robt. Tannahil]
Vocal Library, London: John Souter. 1818
Encyclopedian of Comic Songs, London, 1819. Folger. wo/music
The Blackbird, N.Y.: Christian Brown. 1825. wo/music. LCMD 
  [others of this title earlier and later in LCMD].
Dandy's Companion, np, nd. [American, c 1825?] LCMD.
Newcastle Song Book. Newcastle. W & T. Fordyce, n.d. [1835-40] 
  wo/music. LCMD. 
A selection of the most favourite Scots Songs I, A Selection of 
  original Scots Songs, II and III. London, Wm. Napier 1790-94. 
  LCMD, but lost?

Not yet seen, and mostly without music:

Vocal Melody, by T. A. Arne. LCRB
England's Glory. 1762. LCRB Songs, English texts, according to 
  catalog card.
Percy, John, A Collection of Songs. Folger.
Chaplet. J. Walsh. 1738. LCMD.
The Musical Century, Henry Cary, 1739. LCMD.
Musical Museum, The (London?), 1760. LCMD, but I can't relocate 
  catalog card in title catalog. [Some are alphabetically out of
  order in LCMD title catalog, like Philomel.]
Delicate Songster. M1619 .D32
Bottle Companion, I. Walsh c 1710.
Musical Magazine, London 1767? M1619 .M94
Paddy's Resource, (American). 179x [Card in LCMD, but book in
Vocal Muse, Philla, 1792 LC [Card in LCMD, but book in LCRB?]
Nightingale, Glasgow, 1806. LCMD 
Nightingale. London. 18xx. LCMD M1619.N65.
Crosby's English musical repository. 1807
Crosby's Caledonain Musical Repository. LCMD, but where is card?
Virginia Nightingale, Alex & F'burg, 1807 [mostly Irish] LCRB
New Myrtle and Vine, London, Tho. Tegg. 1806. LCMD
  1810. M1619.S7C5. [Can'f find]
New Musical and Vocal Cabinet. London n.d. H. Gray. LCMD
The Harp of Erin, (American), LCRB.
Myrtle and Vine, Wilson. LCMD. 
Rose Shamrock and Thistle, N.Y. 18--. LCMD.
Musical Magazine, 1767. LCMD M1619.M94 
New Songs, R. Bremner, c 1770. M1619.Mxxx
New Musical and Vocal Cabinet, 18-- Eng,Scot &Ir.
Nobody coming to Marry Me. Dublin, E. Lee M1619.A2N [Book or 
  single sheet song? Song, based on one in Tea Table Miscellany,
  it is in other collections around 1800-1810]
A selection of Scots, English and Irish Songs, Edinburgh: D. 
  Robertson, c 1819. LCMD?
The Patriot's Vocal Miscellany, Dublin, 1804. LCMD, but they 
can't find it. [forgot to see if transfered to LCRB]
British Melody, Or Vocal Magazine, Walsh, 1739 [Title cat.]
British Miscellany, Walsh, c 1703 [Title Cat.]
Collection of the Choicest songs, Walsh, c 1703. [Title cat.]

Manuscript popular music: (by no means complete)  
M1 .M595. Vol. III? Scots and Irish Music, Vol Vol. 1-4
English in catalogue, M1619.M
M1 .A15 vol. 90 Am. and Eng. vocal music c1794-1800.
MS Music Book, 1741. M1 .A1M [plus others seen, e.g., Henry Beck
flute MS, 1786, and a few others]  
M1.A1 is a catch-all where short items are stored alphabetically
by title.

LCMD. Title catalogue - '137 old ballads' is actually, for the
most part, a collection of single sheet songs with music, with
one or two 4 page musical scores, and a MS copy of the popular
18th century Irish tune "Sheila na Guira". The Folger Shakespeare
Library and Harvard have excellent collections of single sheet
songs with music, but I haven't see any of Harvard's except for
the few that have been reprinted.