Up until recently, Trek players only had two choices if they wanted a pet. A non-combat targ for Klingons and some type of tribble for the Feds. While you could only get a targ from the C-Store, just about anybody can breed the non-rare tribbles. And, of course, there is the Cryo Tribble you can breed if you complete a certain mission in the Deferi story arc.
NOW, the C-Store has added both the Borg and Fluidic Tribbles (usable by either faction) and most recently the opportunity to own your very own non-combat pet Sehlat cub! (also available for both factions)
Personally, I'm waiting for my very own pet Exocomp. ;)