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I was wondering what are the Killing vectors of BTZ black hole and how to guess them easily? Will it be the same as of AdS? What then will be Killing vectors for AdS-Schwarzschild e.g.?

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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.

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