The other answers have done a good job of covering the fact that we don't know if singularities actually physically exist, they might represent a breakdown of our theory, etc.
Here's another point which I think is worth making. Within the theoretical and mathematical framework of general relativity, singularities are guaranteed by the singularity theorems of Hawking and Penrose. Roughly, these theorems require the following things to hold:
-The spacetime you're working with satisfies certain energy conditions (we can use very weak ones to prove the singularity theorems, they're pretty physical and not very stringent).
-The spacetime satisfies certain global causality conditions (essentially guaranteeing no closed timelike curves allow time travel to the past).
-There exists a surface in the spacetime from which we could emit light and it could not escape to infinity (a formal way of testing that our spacetime contains a black hole).
If these things hold, the theorems guarantee that a singularity exists! So within the context of general relativity, singularities are inevitable. This was not obvious at first, and people worried that the high symmetry of our exact black hole models, and that maybe realistic black hole collapses might not have singularities. The singularity theorems well us that this is not an out. Very symmetric models do not explain away singularities, the conditions for the theorems to hold are very flexible.
Now, this guarantee requires that general relativity be a final theory which won't fail in the sort of regimes where singularities form. We think general relativity will probably have to be modified into a full theory of quantum gravity to make real predictions inside black holes, but general relativity taken literally requires singularities. It gives us reason to suspect that even if singularities don't survive in the full theory, they almost do in the sense that there will be extremely dense regions in the interiors of black holes.