Do you mean literally sheet music Jim, or printed sources generally. Dick in his The Songs of Robert Burns, 1903 prints The Secret Kiss as the melody (quoting SMM as his source).

The notes in that book also have a list of tunes that were considered for it:

"The tune, by James Oswald, is in Universal Harmony, 1745, 108, entitled The stolen Kiss; in the Caledonian Pocket Companion, 1752, iv. 23, The secret Kiss. Burns was not quite satisfied with his choice of a melody, for in September, i793, he suggested to George Thomson that as it precisely suited the measure of the air, Waes my heart that we should sunder, he might set it to this. Thomson did not act on the advice, but printed it to The old highland laddie, which subsequent compilers have adopted. Burns's alternative melody, Waes my heart that we should sunder, is a characteristic tune printed in Original Scotch Tunes, 1700 also in the Orpheus Caledonius, 1725, No. 9."

I had a look on youtube, but early recording are hard to come by, though there is one from 1907? by Archie Anderson (baritone), Gae bring to me a pint o’ wine, though he uses the more common tune. According to Sheena Wellington (quoted on Mainly Norfolk - The Silver Tassie, this originated in the late 19th century.

Mick