Grebe

Net it, carry it home, turn it into cage furniture. That is the entire loop. Once it goes into the cage it is stuck there unless you shimmer the furniture apart, so decide where the display is going before you craft.

It only appears on desert water sitting directly on sand, so a normal desert trip will never produce one. You have to find the oasis itself. It has almost no health and takes damage from anything nearby, including a stray swing from you, so put the net in your hand before you get close and leave the weapon on the hotbar. Run your luck into the negative and it turns hostile, which is more of a curiosity than a threat.