Where do black holes lead to?

I'm just curious - do they end in a singularity or do they act as wormholes where "white holes" appear in another universe or dimension? Also, if black holes do indeed end in a singularity (1-dimensional point), then what happens to the other 2 spatial dimensions!? Surely everything IN our universe (including black holes) should have 3 spatial dimensions, so how can a 3D object lose 2 of these dimensions? I mean, even if an object were extremely flat, one of its dimensions could still measure 0.0000000000001 Planck units, or would that be impossible due to quantum physics?