Certain Breeds have no icon. So it's not the rarity that makes the icon appear, it's the breed. The icon it'self doesn't indicate how "good" the pet is. Different ppl will argue that such-and-such pet is better in this breed, and some will say the opposite. So really it's up to you which breed YOU want (some pets come in more breeds then others).
However, rarity (being poor, common, uncommon, and rare) makes a bigger difference than breed.
Hope this helps!