I would be fascinated to see how something like a sugar glider or flying squirrel would do in microgravity. Tiny, omnivorous, and can be kept as pets. I wouldn't be surprised if they could eventually figure out how to use their gliding membranes to fly/swim/otherwise move around.
Cats, dogs, mice or rabbit as pets in zero gravity would be a problem, training them to use a zero gravity toilet would be at least very difficult if not impossible. Similar problems with eating and drinking.
Logistics sets some applicable limits:
Animals require food, produce waste. The larger the animals, the more resources required. In an environment where every gram is expensive, this means animals should be small.
Something with a will on it's own moving among critical equipment is hard to plan for. Animals should therefore be confined to their own ...