One thing that I'd like to see is the ability to have saved or named filters. For instance, I use:
Collected - include
Collected (rare) - exclude
Collected (duplicates) - exclude
Pet battle - exclude
Can do battle - include
to find the pets that I want to upgrade with battle stones. I'd love to be able to save this filter set so that I don't have to remember exactly this set every time I get a stone. Perhaps I could call this my "upgrade" filter, and it should then be possible to further adjust the results, so that I can add pet family for a specific family stone, or remove the "can do battle" if I've already upgrade all of the battle pets.
I'd also love to be able to see which of my pets is a favorite (this info is on the armory), and be able to filter on this. Also perhaps a little breed badge on the pet image for pets I own (like a little PP, SB, HH, etc. in a circle stamped in a corner).
I guess I'll stop now.