Besides the famous Yankee Doodle (which has a LOT more verses---and versions---than I think most people are aware of), one of the most popular songs sung by American troops during the Revolutionary War was a song called Chester (this being the name of the melody, and having nothing to do with the lyrics.)

You didn't ask about the War of 1812, but here's a popular song from that war anyway: Ye Parliament Of England. Can't find any decent versions on youtube, though there are a couple of less than great ones if you want to go searching. (I have sheet music for this song in a book, which is how I know that the versions on youtube are not great.)

As for the Civil War, there were a lot of popular songs, both sung and played, which you can most likely find collected on CD at your local library---every library I've ever been to has has at least one album of Civil War songs. But I'll give you a few here as well:

When Johnny Comes Marching Home

a capella sung version

played as a military march, no singing

The most popular Southern anthem (more popular among the Southern side than the official Confederate anthem "God Save the South" was, actually): Dixie

Most popular song with the Union side: Battle Hymn of the Republic

Finally, fiddle tunes, not necessarily sung to, were very popular. Here are just a few of the many that we know were played.