First off, I would have to say that if we take this question on purely empirical and/or logical grounds, there are no a priori reasons to conclude true singularities must be "impossible" or "cannot exist". In particular, with specific regard to the case of black holes, we cannot directly observe anything beyond the event horizon essentially by definition, for an event horizon is crossable in precisely one direction: inward, in this case. And unless there is some sort of radical error in our physical theories such that in reality it turns out to be possible to send information back out across a horizon from a probe within it and in a practical manner, then this is how it will always be. Any probe sent through such would either render its findings forever lost, or if the information could be extracted, it would be intractably difficult to do so. Moreover, even if we did have such a method, from a purely practical point of view, getting a probe to a black hole is all as difficult as interstellar travel, because all extant black holes we know of are essentially dead stars, or the very massive ones located in galaxy cores. None of these are anywhere even remotely close by - which is, of course, a good thing, as even though it is quite difficult to get "sucked" into a black hole as some might understand it (since they don't actually "suck", they gravitate), the environment near one, due to highly energized matter often being present, would be potentially lethal to life.
Thus we can say we do not have any direct evidence, and likely will not, to actually rule on a singularity as being impossible. That said, if we have a theory of black holes which predicts something different under some circumstance for what goes on outside an event horizon than general relativity does, then we might have a shot at being able to at least make a reasoned guess that whatever that theory says happens on the inside is likely to be true: but even then, there could be potentially infinitely many theories that give the same results with dramatically different interiors.
Mathematically, and logically, there is nothing strictly speaking "wrong" with singularities in that mathematical structures including them have to be logically inconsistent or incoherent (meaning: contain a contradiction, i.e. that you can deduce the simultaneous truth of both some proposition and its negation) provided you treat the behavior of the singularity encounter suitably well (and General Relativity actually does this entirely consistently - any particle that ends at the singularity of a black hole ceases to exist, meaning its worldline terminates, mathematically essentially exactly like how a finite line segment in Euclidean geometry has ends), and thus we can take them into the spectrum of logically possible worlds and so it is from this point of view as well, entirely possible one of those worlds could be our world.
That said, there are usually some reasons given to why we might not want to expect singularities. The common one is that in addition to general relativity, we also have another theory of physics: quantum mechanics, and it doesn't like singularities. In particular, from at least a naive point of view, confining particles to an exact location (since in quantum mechanics, physical parameters of a system like location have bounded information), i.e. the zero-thickness singularity point, effectively drives their energy to infinity, and yet an infinite amount of energy leads to worse problems than an infinite amount of spacetime curvature. More seriously, the mathematics of quantum field theories - the theories we use to incorporate relativistic effects and dynamic creation/destruction of particles into quantum mechanics - fail to accommodate general relativity entirely. The usual way this is handled is to assume that general relativity must then give out, and that it will be suitably modified by quantum mechanics. However, one might argue that if we are going to assume one or the other theory gives way, it might be that instead it is quantum mechanics which is the one that dies and general relativity, or some extension thereof, is the really more fundamental theory, and if that is the case, then we could indeed have a true singularity at the black hole center. Going even further, one can stake a case that in fact we don't really even understand quantum field theory well enough and thus it may be premature to speculate about interactions and/or conflicts with general relativity.
So the answer is: actually, we don't know. They might be out there, and an honest investigator would be entirely open to this possibility. Then again, they also might not be. We should keep an open mind, and keep investigating - ideally, investigating things with methods "outside the box" compared to what we have been doing so far, including spending more money on constructing ever-larger particle accelerators, as the results from the last one, the Large Hadron Collider, aren't looking too promising insofar as providing any solid (>4 sigma) leads on new physical processes which cannot be accounted for by existing theory are concerned. (Sabine Hossenfelder published an article about this recently, though I don't have a link.) Einstein supposedly(!) said that to try something over and over - which here would mean to build a bigger and bigger machine - while expecting a different result, is idiocy. (Whether he did or not, I'd find that a good point nonetheless.)
