I've put together a guide to catching Tiny Carp, based on what has been discovered so far. I'd post a link, but forum won't let me.
Tiny Carp are bound when caught, and when learnt unique and non-tradeable, so the only way to gain them is to catch them. I reckon they are are rarest minipet caught by fishing. When first added to the test realm they were fairly easy to catch. But I've ploughed through 14,000 catches since then, and not caught one.
Given their rarity, it isn't possible to confirm all possible catch locations. Indeed, there are some oddities, that raise the possibility of certain pool types containing more than one type of Tiny Carp, which could change the strategy for catching them. That should become clearer once more people report catches when the patch goes live.
There's a quirk with the Tiny Blue Carp, whose stats are always as good as, or better, than the other Tiny Carp. At level 1, the Tiny Blue Carp's health is 155 (compared to 151 or 152 for other Tiny Carp) and their attack is 12 (compared to 11), but their speed is still 12 (the same as the other Tiny Carp). Abilities vary a bit, but are broadly similar. Since all are rare quality, I'd expected the stats to be balanced equally. Can anyone explain this?