For lack of a better term, I use the word filters. I think it would be extremely useful to implement some type of sorting option for the stables. By this, I'm looking for a way to sort the pets stored within the stables, not the stables themselves. For those of us with multiple stables, it can be extremely time-consuming and/or frustrating to rearrange our stables to our liking. I've thought of a few different types of sorting options and listed them below.

Sort by:
species
level
rarity
themes
number of traits
age - either by age or age level such as adolescent and adult
able to breed - note this being useful due to the filter of fertility
able to be mutated
skills - agility, dexterity, intelligence, health, and strength

I've also thought that maybe these categories could be used as a stable search function, specifically to our own stables only to find useful pets, like those that may be used in breeding projects, or battle setups.

I've seen something like this suggested before and I believe it was said that it can't be done with the way the stables are set up right now. Most of this can be done with advanced search, so that's your best bet.. along with obsessive organization hah.