Naked singularity Many consider naked singularities as a fundamental problem and that it should be always covered by a horizon (Cosmic censorship hypothesis). But why naked singularities are really a problem?
If we consider electrodynamics and the Coulomb potential, we have a singularity at $r=0$ but quantum electrodynamics solves the problem. 
General relativity being a classical theory we have also a singularity and with the hope that quantum gravity will remove it. But we don't need necessarily a horizon. 
At classical level, naked singularities should not scare us. Why are they always disregarded?
Edit after the answer by John Rennie:
Thanks John Rennie for your answer. But we actually express exactly the same thing. I'm not saying that a singularity is not a problem, of course it is. But that singularity is not a problem within a classical theory because we expect or hope that the problem will be solved in the quantum regime. 
Going back to my previous example, classical electrodynamics, no-one tries to hide a singularity behind a horizon but in general relativity we try to solve the singularity problem within the classical theory. The "Cosmic censorship hypothesis" tries to "solve" (most exactly hide) the problem within the classical regime.
My question then is, why not thinking that a naked singularity is a fair solution in the classical theory but the singularity would disappear in the quantum regime without necessarily imposing a horizon in the classical theory. 
 A: Your initial assumption is wrong. Classical singularities do scare us, but we have a resolution for that problem because quantum mechanics modifies the classical behaviour at short distances. For general relativity no such safety net currently exists, though most of us believe quantum mechanics will remove the singularities in GR as well.
It is a basic requirement of a theory that if we know the state of a system then our theory can predict its future evolution. Technically this property of a theory is called global hyperbolicity. If a theory is not globally hyperbolic then causality breaks down because we cannot predict what cause will have what effect.
The problem with a singularity is that while we can calculate the trajectory of an infalling particle up to the moment it reaches the singularity we cannot predict what happens at the singularity or for any later time. This happens because the curvature tends to infinity as we approach the singularity and we can't do arithmetic with infinity.
However provided the singularity is hidden behind an event horizon the unpredictability doesn't matter because everything behind the horizon is causally disconnected from us - the unpredictability can never affect anything that we can observe. But if the singularity is not behind a horizon, i.e. it is naked, then what happens there can and will affect us. That means our theory (GR) is no longer globally hyperbolic and therefore cannot predict the future. We're in trouble!
This also happens in classical physics, and you use the example of the Coulomb potential. If we consider a positive and negative charge on a direct collision course then their equations of motion also become singular when the distance between them falls to zero, and there is no way to calculate what happens afterwards. But of course we know that we have to resort to quantum field theory at very short distances, and this removes the singularity. Panic over.
The problem is that quantum mechanics does not (currently) come to our rescue in GR because we have no theory of quantum gravity. As I mentioned at the start, I doubt you'd find a theoretical physicist who really believes singularities exist - we all think some form of quantum effect will remove them. But this is currently only wishful thinking and there is zero evidence to support it.
There is one final point to be made about causality. GR is time symmetric, and that means if we cannot predict what happens for particles hitting a singularity that also means we cannot predict what comes out of the singularity. If we observed a naked singularity we simply could not tell what it would do next.
A: It's also worth noting that black hole solutions with naked singularities tend to have other pathologies, like closed timelike curves, that don't directly involve the singularity, and even if you removed the singularity with a patch, these other pathologies would remain.
