Yes, "singularity" does exist.

Nothing travels, as such, any closer to the speed of light than (+/-)186,000mps ((+/-)300,000kps). It can only be closed-systematically accelerated to. Accelerated to, as in accelerating into a point (into a "singularity"). Einstein's fourth dimension of space, time, is a point (0-point). That point is what merges space and time to a space-time point-bubble elasticity. A finite *potential* of infinity observed outside in. The fulfillment (of infinity) inside out.

A black hole is, probably, entirely its 'doughnut' torus and/or ring(s) of event horizon. It isn't a sphere with an atomic core like our star or any other similar spherical star. You, if you were a traveler, were to go with the grain of a black hole's event horizon you might probably be going with the grain of an accelerating expansion (something like 'c' squaring) that could, ultimately, just keep on expanding to infinity. But between beginning to go with the grain, and going for infinity, you the traveler might encounter an infinity of territory -- an infinity of "well" and frontiers, so to speak -- opening in 'horizons' and having to be crossed. Survive, you the traveler might want to stop along the way to find what you were searching for in the first place . . . because your odds of getting back home would be one point-horizon in an infinity of point-horizons.