Worldwide they've elected dogs, cats, a brown mule, a beer-drinking goat, a rhino, a monkey, a hog.
Some specifics
Rabbit Hash, Kentucky
Elected a dog as mayor 4 times
Milton, Washington
Elected Boston Curtis, a brown mule, as the new Republican committeeman
Lajitas, Texas
Elected Clay Henry, a beer-drinking goat, as Mayor. When he died his son Clay Jr. took over.
Talkeetna, Alaska
Stubbs, the cat, has been Mayor since 1997