Songs featuring felines or canines

Do the Dog, by Martin Mull
The entire Snoop Dogg back catalog.
The Stooges "I Wanna Be Your Dog"
Duh!
"Drive By" - Glen Phillips (Intro: "I was the driver for the drive-by of the neighbor's dog"; Chorus: "And I prayed, "Dear God, if You save this dog, I will never get high, I will never jack off, I will do all the things that I should but have not, I'll be a good boy from now on" ")
Pads, Paws & Claws, Elvis Costello (Paul McCartney)
Can Your Pussy Do The Dog – Cramps

Hah! Two in one.
Do the Dog – The Specials

Not the Donkey…
Stray Cat Strut - The Stray Cats
Golden Retriever by Super Furry Animals
Doggystyle by Archigram (it samples The Stooges)
Grandaddy, Stray Dog & the Chocolate Shake
Pink Floyd, Dogs
"Ship of Gold" - Clutch
Walkin' the Dog - Rufus Thomas (and many covers)

Move It On Over - George Thorogood

Dog and Butterfly - Heart (eeeewwwww)

Dawg-Ola - David Grisman (and similar Grisman tunes)
Stray Cat Blues - Rolling Stones

Year of the Cat - Al Stewart

Cat Scratch Fever - Ted Nugent

China Cat Sunflower - Grateful Dead
Cat Claw - The Kills
Riders on the Storm, by The Doors
Dog On Wheels - Belle & Sebastian
Kitten - Andy Votel featuring Jane Weaver
"Dogs of Lust" by The The
"How to Skin a Cat" by Hüsker Dü