The Scots Musical Museum version (that is, the original) is sung by Tony Cuff on The Complete Songs of Robert Burns, vol. 1 (Linn Records CKD 047), which is also on Auld Lang Syne: A Fine Selection of Popular Robert Burns Songs (Linn CKD 088). Sound clip is HERE. The tune is in the following as well as in Scots Musical Museum, Dick's Songs of Robert Burns and Kinsley's Poems and Songs of Robert Burns:

C.V. Stanford, The National Song Book (Boosey, 1907, p. 72; as "Afton Water")
Maver's Collection of Genuine Scottish Melodies (Robert Maver, n.d., p. 38 [tune name: "The Brae of Ballenden"]
Helen Hopekirk, Seventy Scottish Songs (1905; Dover, 1992, pp. 42-45)
A Song of Scotland (Wise Publications, n.d., pp. 74-75; as "Afton Water")

    I don't believe the "familiar tune" was composed by Alexander Hume, though Margaret Boni (in Fireside Book of Folk Songs, pp. 106-107) and William Cole (in Folk Songs of England, Ireland, Scotland & Wales, pp. 135-137) attributed the tune to Hume. According to The Book of Worl-Famous Music by James J. Fuld (4th ed., p. 228), the version ("known particularly in America") was composed by J[onathan]. E[dwards]. Spilman, who "was born in 1812 in Greenville, Ky., became a lawyer, then a minister, and died in 1896 in Flora, Ill."

The first edition at the Levy collection is:

Title: Flow Gently Sweet Afton. A Ballad. [no images given]
Composer, Lyricist, Arranger: Written by Robert Burns. Music Composed and Arranged by J.E. Spilman.
J. E. Spilman Publication: Philadelphia: George Willig, 171 Chesnut[sic] Street, 1838 [Sold at P.H. Taylor's Music Store, opposite the Banks, Richmond, Va.]..

See other editions:

Title: Flow Gently Sweet Afton. A Ballad. [only one page is given; link to cover is inactive]
Composer, Lyricist, Arranger: Written by Robert Burns. Music Composed and Arranged by J.E. Spilman.
J. E. Spilman Publication: Philadelphia: George Willig, 171 Chesnut Street, 1838.

At American Memory:

Flow gently, sweet Afton, music by J.E. Spilman (Boston, Massachusetts, Oliver Ditson & Co., 1880) (Another copy)

      There're several compositions by others, which I think are less known.

      The Alexander Hume version has a different melody, which was sung by Joseph Hislop (tenor) and recorded in 1922 as "Afton Water." The recording is on The Star o' Rabbie Burns (Moidart Music Group MIDCD 004), whose notes say "The original setting (1791) has long been superseded by this typically Victorian version of 1855 by Alexander Hume."

~Masato