Black hole with two singularities?

There are only four known stable black hole geometries: Schwarzschild, Reissner-Nordstrom, Kerr and Kerr-Newman. We expect that any random assemblage of matter dense enough to form a black hole will relax into one of these four geometries by emission of gravitational waves. None of these geometries has two distinct singularities, so (as far as we know) it is impossible to have a black hole with two distinct singularities.

You suggest possible arrangements of matter in your question, and as Jerry says in his answer these could occur as transient states in a black hole collision. However in time (actually very quickly :-) they will relax into one of the four known geometries with just a single singularity.

I think the closest model to what you're talking about would be two colliding black holes, during the intermediate period where their horizons had merged, but the central objects had not yet collided. These systems are very different from isolated black holes, as they give off significant gravitational radiation, and have horizons that rapidly change in shape as they give off this radiation.