I wanted an easy way to keep track of my upgraded pets as well as the ones I plan to upgrade later. The Flawless Battle-Stones are obtained through pet tamer battles while the odd random family ones collected through successful wild battles. Note that winning against a higher enemy team gives you a better chance at a stone.
If your bag space can handle it, I recommend levelling your green quality pets to 25 and then using the stone so you’ll get credit for levelling the same pet twice – particularly if you’re going for Pro Pet Mob.
Once you have the achievement, don’t be afraid to use your stone on your uncommon pets (or lower) ASAP for easier levelling.
As a guide, use a stone on any pets you can’t upgrade another way, like quest reward and argent pets or low spawn pets below.
Baby Blizzard Bear
Tranquil Mechanical Yeti
Personal World Destroyer