"Jerusalem Cuckoo" (aka. "Donkey Driver")—originally a music hall song, but it's entered the folk tradition.

"Lie Stone Dead in the Market" (The Donkey Song)

"Doran's Ass"

"Whoa, Mule, Whoa"

"My Little Brown Mule", by N. Howard "Jack" Thorp, 1912, about his pet trick mule

"The Old Gray Mule"

"Old Matha Gummersal's Mule" (see the Yorkshire Garland Group website)

"Go [A]long Mule"

"The Mule Song", by Edward Harrigan and Dave Braham, pub. 1882

"The Muleteer", minstrel song by D.A. O'Meara

"On the Road to Cook's Peak"

"A Man Named Hods"—his mules survived an Indian raid better than he did, but the song is obviously more about the man.

"There's a Mule Up in Tombstone, Arizona", a composed song Burl Ives used to sing.

"Mules": On mules we find two legs behind...

"Hacia Belén va una burra", a villancico initially about the burro the Virgin Mary rode to Bethlehem (and did you know Mary and Joseph were chocolatiers?!)

"Le mule de San Giacomo"

? "My Sweetheart's the Mule in the Mines"

? "Compré una mula en Tafalla" (I Bought a Mule in Tafalla)

? "Quita la mula rucia" (Unhitch The Light-Brown Mule)

"Creeping Jane", about a racehorse. Surely mentioned in the horses thread, but just in case.

"Galtee Farmer" and related songs, about a farmer and his son who sell their old horse to a slick operator who tarts it up and sells it back to them for a hefty profit.

"Chilao", by Herbert Henry Knibbs

You might also search for "moke", a slang term for a donkey (Brit.) or horse (Oz/NZ).