A gravitational singularity is not a mathematical improbability, it's a mathematical inevitability: The numbers say that any mass compressed to a sufficient density will collapse into a singularity.

As to whether or not they actually exist, the numbers also say that singularities exist at the centers of black holes, and black holes do exist (at least, there are objects so small and massive that according to the numbers they can't be anything else).

Do singularities annihilate rationality? Yes, by definition: At a singularity, the formulas used to describe and predict conditions in the rest of the universe become useless (dimensions shrink to zero, density and gravity become infinite, etc.). A singularity is literally indescribable in classical terms.
Source(s):
http://en.wikipedia.org/wiki/Gravitational_singularity
http://en.wikipedia.org/wiki/Schwarzschild_metric

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