Even if we had no clues in the journals to tell us that the men occasionally sang popular songs of their time, we would suspect it because in those days, before vicarious participation in athletics became a measure of maleness, singing was a "guy thing" both in Europe and America. But musical activities are mentioned more than thirty times in the journals, and singing is specifically cited a number of times.
So what songs might they have sung at the conclusion of that traditional holiday ceremony, following the "Selute" and the "Shoute"?
It's easy to list some of the songs the men could not have sung because they had not yet been written. Those would include "Silent Night" (1818); "O Little Town of Bethlehem" (1874); "Angels we have heard on high" (1855); "Away in a manger" (1855); and "What child is this?" (1871), though the tune was popular in 1803.
A few that the men might have sung, because they were more or less well known around 1800, include: "O come, all ye faithful," but perhaps in 3/4 time; "The first Nowell" or "God rest you merry, gentlemen"; "Away in a manger"; "While shepherds watched their flocks by night"; or "All hail the power of Jesus' name."