The geometry of the BTZ black hole, including its Killing vectors, is described in Steve Carlip's paper (which I just noticed is also referenced by the wiki article). Another good reference on this topic is this one.
Basically, in the Schwarzschild-like coordinates, the Killing vectors are just $\partial_t$ and $\partial_{\phi}$.
As for how these Killing vectors can be "guessed" - the BTZ hole is obtained by quotienting AdS space by an isometry. Since the isometry is itself generated by a Killing vector, I'd start by looking for Killing vectors on the original AdS which commute with the chosen isometry-generating one.
As for AdS-Schwarzschild, I'm not sure - it's only asymptotically AdS, so the AdS Killing vectors would only apply in some sort of limit.
This post imported from StackExchange Physics at 2014-03-22 16:56 (UCT), posted by SE-user twistor59