Who is at fault if car 3 runs into the back of car 2 because car 2 has crashed into the back of car 1?

Car 3 is At Fault for striking car 2.

If car 2 and car 1 are sitting stationary (e.g.: at a traffic light) then car 3 is responsible for entire accident

The charge may vary from jurisdiction to jurisdiction but it will probably be some variation of; "following too closely" - "failing to have vehicle under control" - "failing to give full time and attention" etc.

If moving in traffic, car 3 (and car 2) should have been maintaining a safe following distance from cars in front of them in order to avoid the collision.

You are at fault if you are car three as I surmise.