Thanks for all of the wonderful insights!
Gandalf is a cat!?! Ouch!
Anyway, just wanted to add that I know of two citations of dogs and cats in Tolkien's world and so will build a case using those (and will exclude any contrary evidence

).
Huan - This dog is the companion of Luthien and Beren during their quest for the silmarils. It not only kills the Red Maw, but also defeats Sauron. Pretty impressive canine resume.
The cats of Queen Berúthiel - These ten cats were used to spy on others; the nine black ones spied upon the populace, and the white one kept watch on the nine (as you just can't trust them!

). Last seen sailing southward on a boat with the banished Queen, who was as loved as her pets. Surely these felines loved and were loyal to their Queen, yet everyone else feared and despised them.
Need I say more?
Anyway, if I were to post a poll, which I won't, and I were to ask which pet the One Ring were more like, and I were to restrict the poll to 'dog' or 'cat' (which I wouldn't, seeing how most polls require much consideration, meaning that I would have to list each and every animal known to exist or to have ever once existed, and even including fantasy creatures
plus the 'other' and 'all of the above options,' the poll still wouldn't be fit for purpose...), which would you choose based on the information regarding these pets as seen above?