When the Argent Tournament appeared, I got excited at the pets each faction offered. For Alliance, I had access to both the Teldrassil Sproutling and the Dun Morogh Cub. While my warlock jousted eternally for a Crusader title, my druid was content to just snag the sproutling and the cub. Whenever she out being a bear, she'd have the sproutling out; whenever she was throwing heals, she'd have the cub running by her side.
Now that I have a second druid to cover the caster specs, my bear druid is a full-time feral. The cub doesn't come out to play anymore, but the Teldrassil Sproutling dances and sleeps just about anywhere.
On my lowbie Horde toons, I always get a Horde Balloon. There's always a Celestial Dragon by my guild's bear tank when she runs in for the pull. My caster druid has the mechanical bunny out, since she's an engineer who loves cute animals like Mylune. One of our moonkin hopes that non-combat pets will be nameable in Mists so he can name his always-present Moonkin Hatchling after his daughter. D'awwww!
With Pet Battles coming out, there will always be people hunting for the next best performing pet. But do you have any pets you're already partial to on certain characters for other reasons?