The skills I can think of are the advanced magics, windmill, ranged.
But these skills are always on that one pet that everyone gets, and then ends up never using for more than bashing/storage.
Pet's aren't just healers and storage, they are stuns, bashers, and can also hold monsters with counter AIs.
But the only skills I need from them are heal (though I hardly use), first aid, counter, and ice/fire/light bolt.
Magic mastery will just make them better healers.
Sharp mind I'd like, every time my pet is attacking an enemy and I try to smash it, I never see if it has a skill loaded or anything.
I've often run into a counter or windmill when this has happened.
and more HP
No one really heals their pets anyways, my 400 HP bear dies as fast as my 50 HP dog.
These will all be on different pets so re birthing would be very useful!
Rebirthing pets... I'd rather not.
My favorite special pets are the ones with special non-skill abilities.
For example, the snake with its poison and the herb pets.