It is quite likely that singularities are artefacts resulting from the fact that Einstein’s theory does not take quantum effects into account, and that they will be absent in a more complete theory of quantum gravity. Yet even if you leave aside quantum theory, and stay strictly within the framework of Einstein’s theory, it is likely that most singularities are, if not absent, then at least well-concealed:
The hypothesis of cosmic censorship states that, whenever a body collapses so completely as to result in the formation of a singularity, a black hole will be formed so that the singularity will be hidden behind the horizon, and thus completely unobservable for anyone outside the black hole.
At the present time, this hypothesis is unproven. Indeed, there are some counterexamples, but they describe idealized situations which are not likely to tell us anything about the real world. Finding a proof that, for all realistic collapse situations, there is indeed cosmic censorship, is one of the great open problems of general relativity research.