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Good question! Singularities are really mathematical features and not quite physical. They are essentially coordinates where physical quantities become infinite, e.g. the spatial curvature of space becomes infinite at the center of a black hole.
It's important to remember that while math is the language of physics, and math can often lead to a better understanding of and even new discoveries in physics, there isn't always an exact one-to-one correspondence. In this case, is there really a region of infinite curvature at the center of a black hole, and did the universe begin as a singularity of infinite density? It's tough to say because as both curvature and energy density become very large, near what's called the Planck scale, our theories (General Relativity and the Standard Model respectively) start to break down.
There's nothing here!