Since you mentioned Rufus Thomas:
Since you mentioned Rufus Thomas:Rufus Thomas, "The Dog"
Meanwhile . . .
Talking Heads, "Animals"
Stick McGhee, "One Monkey Don't Stop No Show"
Major Lance, "The Monkey Time"
The Graham Bond Organisation, "Camels and Elephants"
Blind Lemon Jefferson, "Rabbit Foot Blues"
Jefferson Airplane, "White Rabbit"
La Vern Baker, "Gimme a Pigfoot"
The Beach Boys, "Mrs. O'Leary's Cow"
Willie Dixon with the Big Three Trio, "I Ain't Gonna Be Your Monkey Man No More"