The pet images presented below are taken from warcraftpets.com. The pet names serve as links to the pets respective pages at warcraftpets.com so you can look in greater depthy at their abilities if you wish.
Contenders...I've based my selection on the 12 animals associated with Chinese New Year.
Rat - Giant Sewer Rat
There's numerous rat pets available, all sharing the same set of abilities, but my favourite is the Giant Sewer Rat. I really like the fact that it's giant and being a fishing-fan had two in my collection (I've since given my second away). He's from the critter family and has largely critter abilities, with only one being non-critter (beast).
Ox - Clefthoof Runt
There's no Ox pet in-game so I've gone with the Clefthoof Runt which I think is the closest to it (and one of my favourite looking pets). Clefthoof Runt's are from the beast family and have a combintaion of beast and critter abilities (ration 4:2).
Tiger - Winterspring Cub
Whilst there's no pets in game that look like a real tiger (Panthera tigris) there is a number of big cat models (mostly in the form of cubs). Only three pets are named specifically as tigers, the Jade Tiger / Spectral Tiger Cub / Zipao Tiger and these pets are from promotions / TCG loot so I've chosen to discount them. Instead I've gone for the Winterspring Cub, which is purchasable in-game, has entirely beast abilities.
Rabbit - Darkmoon Rabbit
Again, theres numerous rabbit models in-game and I've opted the Darkmoon Rabbit. After all who's to say that it's blood around it's mouth and not sweet and sour sauce!?! This critter has a variety of abilities including three non-critter ones spread throughout the three slots allowing real ability-flexibility.
Dragon - Mechanical Pandaren Dragonling
Mists of Pandaria brought a whole new oriental feel to WoW, including new mounts and pets. It share's it's model with some other hatchling's and is very similar to the Celestial Dragon but is unique in the sense that it's mechanical - which is why I've chosen it. Unsurprisingly it's from the mechanical family, sub-family mechanized critters. What I especially like about this pet is that it has abilities from four different families: dragonkin, elemental, flying and mechanical (x3) making it great for battling a range of pets.
Snake - Zooey Snake
2013 is the year of the snake! The game is filled with various snakes, adders, vipers etc. which share the same abilities. I've chosen the Zooey Snake as I like it's name and skin pattern (above). Snakes are beasts and have mostly beast abilities with one humanoid ability.
Horse - Sea Pony
Not a true horse but who doesn't love the Sea Pony?? It's also a chance to bring an aquatic pet into the mix. All it's moves are aquatic making it an excellent pet to take on elementals. Also has the aquatic heal cleansing rain.
Sheep - Black Lamb
Black Lamb's can be found an captured in Elwynn Forest. They're critters with 100% critter abilities. Has the ability Chew which I personally find quite amsuing.
Monkey - Darkmoon Monkey
If he's good enough for Jeremy Feasel's team then he deserves to be mentioned! I'm sure most of you have encountered Mr. Monkey and his terribly annoying Banana Barrage. It quite often bugs out when I'm fighting him and either deals mental amounts of damage or doesn't stop. He has 5/6 beast abilities with one humanoid ability.
Rooster - Westfall Chicken
Again, there's a few varieties of chicken in-game which share the same abilities. I've gone with the Westfall Chicken which you get for completing the quest CLUCK! It's from the flying family and has mostly flying abilities a couple of critter abilites.
Dog - Core Hound Pup
The Core Hound Pup isn't exactly a dog but I like it. If you don't have one they're easy to get and come free when you get an authenticator. He's one of the four pets in the elemental sub-family of mythicals. Whilst the Core Hound is from the elemental family it only has one elemental ability mixed with three beast, a critter and a humanoid ability - giving it a good variety.
Pig - Mr Wiggles
Mr Wiggles can be acquired as a quest reward during Children's Week and is a very cute little piggy! He's from the critter family and has mostly critter abilities (four in total). He also has an undead and a beast ability. One of his critter abilities is called Buried Treasure which sounds pretty cool.
Having 12 contenders has made choosing a final three quite tricky but it does mean that I've got a range to pick from. I've chosen the Clefthoof Runt because despite my love for it's appearance I've never leveled mine. My second choice is the Mechanical Pandaren Dragonling because I love how the mechanized critters look in general and I could do with leveling another mechanical pet. The Sea Pony is another pet that I love and I worked really hard to get but have never leveled so now is his chance to shine!