Cats: Are purchased from a 'Crazy Cat Lady' just east of Stormwind in Elwynn Forest (44, 53).
White Kitten: Sold by Timmy in Stormwind. He isn't really rare but despawns rather quickly after selling just 1 kitten. If he just doesn't offer you anything for sale then he has already sold his 1 kitten. Try again later.
Owls: Owls are purchased in Darnassus.
Great Horned Owl
Snowshoe Rabbit: Sold in Dun Morogh
Snakes: Snakes are sold from a wandering vendor in the Valley of Spirits in Orgrimmar.
Prarie Dog: Sold in Thunder Bluff.
Cockroach: Sold under the bank in UnderCity.
Ancona Chicken: Sold in the Shimmering Flats in Thousand Needles.
Prairie Chicken: Short quest in Westfall from the chickens around Saldean's farm.
Parrots: Sold in Booty Bay.
Macaws: These parrots are rare drops.
Hyacinth Macaw: found in Stranglethorn Vale
Green Winged Macaw: found in The Deadmines
Mechanical Squirrel and Yeti: Made with Engineering (usable by anyone).
Mechanical Chicken: Quest reward after completing the three beacon quests.
Exotic Pets: These pets are quest rewards from a guy in the Burning Steppes. They can be soloed with the right class but usually require a group in LBRS.
Smolderweb Hatchling (spider)
Dragons whelps: Dragon whelps are rare drops from dragon whelplings.
Tiny Crimson Whelpling: found in Wetlands
Tiny Black Whelpling: found in Badlands
Tiny Emerald Whelpling: found in Swamp of Sorrows (I think)
The Disgusting Oozling is a rare drop from various Oozes around the world. It is unusual due to it aura. It turns the owner green and even lowers some resistances.
Collector's Edition Pets: After loading up your character, you find a present in your inventory that you give to an NPC in exchange for one of these pets.
Mini-Diablo (Lord of Destruction)
Nether Drake (Burning Crusade Expansion Pet)
The was a Murloc pet named "Murky" given out at one of the Blizzard Festivals. You had to redeem a coupon in Ironforge to obtain him.