Harry Chapin, Cat's in the Cradle. A great song with an important message by a songwriter (1942-1981) who died too young.
Bob Dylan, Cat's in the Well
Bent Fabric, Alley Cat
Tokens, The Lion Sleeps Tonight
Rooftop Singers, Tom Cat. From those far-off and fabulous hootenanny days.
Norma Tanega, Walkin' My Cat Named 'Dog.' Tanega was a mid-'60s one-hit wonder. What became of her? Story here.
Kinks, Phenomenal Cat
Lovin' Spoonful, Nashville Cats
Buck Owens, I Got a Tiger By the Tail
Al Stewart, Year of the Cat. HT to Jeff Hodges for reminding me of this one.