Why exactly are singularities avoided or "deleted" in physics? What is the real reason that make us reject singularities everytime we see them in a theory/model?
For example, in GR, it is predicted that black holes singularities have infinite density. This makes GR useless to predict the behaviour of the singularity.
There are some proposed alternatives to study the behaviour of singularities and all of them, assume that singularities do not have infinities, or even, "delete" singularities, replacing them with something different, for example, "tunnels" to another universe (see Black Hole Cosmology https://en.wikipedia.org/wiki/Black_hole_cosmology)
But why does this happen?
I've been told that when we say that singularities "break laws of physics" what we actually mean is that GR does not apply there, but another version of GR or another set of laws different from GR and our standard model would apply there. That physicists avoid singularities (or, at least "infinities") because it would really/actually break all the laws of physics and could not be described by any theory (not GR, neither a different version of GR nor a different model/set of laws of physics than our standard models...etc), and because of that, they model singularities with no "infinities" (or even they do not model/consider any singularity at all) in their works/theories. 
In summary, that singularities (with "infinities") in, for example General Relativity, are avoided/ignored or thought to not exist because if they would actually have infinities, they would really/actually break literally all laws and could not be "governed"/described by any model/theory. For that reason, they are considered to be not real, having finite densities or being replaced by something different.
But is this right? What is the real cause that makes us reject singularities (with "infinities")?
 A: When you ask most working physicists this type of question, the answer you get tends to be an oversimplified one that is partly just based on experience and conservatism. There were singularities and acausal behaviors in the classical electrodynamics of point particles, but this eventually got pretty much cleared up by QED, so the moral that people took to heart was that this was how to look at all singular or acausal behavior, as just a sign that the theory was incomplete. If you look at the work of specialists in relativity, you will find a more more complicated description. A good discussion of this kind of thing is given in Earman, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes, which Earman has made free online. Earman is both a physicist and a philosopher, and the book contains some of each.
We would like to be able to use GR to make predictions. There are some things that we don't expect the theory to be able to predict, such as what the big bang should have looked like, so the feeling is that a past spacelike singularity is OK. This gets formalized in the notion of a globally hyperbolic spacetime. A globally hyperbolic spacetime is one in which we can give initial conditions on a spacelike surface (called a Cauchy surface) and then evolve the field equations forward and backward in time. Basically we have global hyperbolicity if there are no closed, timelike curves and no naked singularities.
A naked, i.e., timelike singularity breaks global hyperbolicity because if you try to draw a Cauchy surface on a spacetime diagram, you can't, because the singularity makes a hole in the topology.
Black hole and white hole singularities don't break global hyperbolicity because they're spacelike singularities that are either in our future or in our past, never both.
If you have timelike singularities, then physics has serious problems with prediction, as shown by Earman's memorable figure and caption below: 


The worry is illustrated in Fig. 3.1 where all sorts of nasty things -- TV sets showing Nixon's 'Checkers' speech, green slime, Japanese horror movie monsters, etc. -- emerge helter-skelter from the singularity.

The reason that GR can't predict what comes out of the naked singularity is simply that we can't even formulate the initial conditions in an appropriate way, because a Cauchy surface doesn't exist.
It's actually quite possible that cosmic censorship fails in realistic gravitational collapse: What is the current status of cosmic censorship? "Quite possible" doesn't mean that I or any physicists I know want to bet a six-pack on it, but that there are serious researchers who think this is a reasonable possibility, and they've suggested actual observations to check. If this turns out to be true, then generically, such scenarios lead to consequences like the release of arbitrary information and infinite energy from the singularity, which is just a polite way of saying green slime and lost socks.
People who don't like this possibility pin their hopes on the fact that there is a Planck scale where we know that GR becomes inconsistent anyway, and definitely needs to be replaced by a theory of quantum gravity. However, I don't know of any argument that convincingly suggests that in a theory of quantum gravity we necessarily recover predictive ability.
A: You have to distinguish between reality and our current physical theories.
When we talk of a "singularity", we more or less mean that we are at our wits end concerning a certain phenomenon. There are two very different alternatives:


*

*A singularity (the physical object), is really nothing special; just some natural phenomenon which can be reasoned about and fully understood - just not with our current version of our theories. It is not "catastrophic" or "mysterious" except that we get a "division by zero" (or other mathematical-technical problems) in our theory, simply because it is incomplete or partially wrong. It is like any other phenomenon which took us years, decades or longer to figure out, and which eventually can be described elegantly by some yet-to-be-discovered theory (say, the Grand Unified GR+QT we are looking for, or something completely new).

*Or: A singularity (the physical object) is actually something really different, a totally new category of reality. Worst case, since we never get inside the envent horizon and thus close enough to actually measure anything about it, we will never, ever, find a fitting theory; or if we by chance or great intelligence do find a correct theory, we might never be physically able to verify it - which means that it is outside of our scientifical process. Remember that even the most advanced theories or thoughts we can come up with are worthless if they are not verifiable.


To answer your question: in the first case, "deleting" a singularity means to fix our theories - it's what physics does best.
In the second case, we're done for. This is of course unsatisfactory, hence people are trying to get around it in whatever way possible.
NB: as pointed out in the comments, nothing of this makes GR or QT obsolete. In the same vein as Newton's is still very much useful for everyday cases, until we find some measurements that directly contradict GR or QT, they are very much "fine" in their respective regimen, and it is well conceivable that if we find a Unified theory, it may be so complicated that the "simpler" ones may stick around afterwards...
A: 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.)
A: In my opinion, singularity is a flaw in the model that will be eliminated sooner or later. In nature, all quantities are finite and their description must be finite. In my opinion, a good method of eliminating the singularity is the use of imaginary values of quantities. Then the real quantities in the singularity are zero, and the quantity remains finite imaginary. The imaginary value means the oscillation of a value with an amplitude equal to the imaginary part. For example, if you calculate the radius of an electron, assuming that its mass is electromagnetic, it is equal to $$r_e=\frac{e^2}{6im_e c^2}=-i\frac{e^2}{6m_e c^2}$$ The real part of the complex value of the quantity means the mean value, and the imaginary part is the standard deviation. This means the oscillation of the size of an electron with an amplitude equal to the imaginary part. Other singularities are destroyed on the same principle. But this is my opinion, it can meet the opposition. Unfortunately, this idea of destroying a singularity only works if space is real. If the space is complex, then the coincidence of the values of the real and imaginary parts of the event is inconceivable, but possible. For this, the space must become imaginary, which must be excluded. In order for the damping or growth of the solution to be impossible when using eigenvalues of energy or momentum, the phase of the complex coordinates must be constant. It turns out that pure imaginary eigenvalues are impossible.
A: Any our theory has limited applications domain. In other words, any theory must be accompanied with numerous inequalities specifying its limitations. These inequalities are often forgotten and we pay the price by obtaining physically meaningless expressions. For example, two charges (electron and positron) interact via the Coulomb potential $1/r$, so the first infinity arises when these two particles merge into something neutral with releasing infinite amount of energy (never observed). This bug is corrected in QED - in a different theory.
Another infinity is the famous UV catastrophe in the black body radiation. It is corrected in QED too.
QED itself is badly formulated and leads to infinite corrections to experimental mass and charge. This is "correced" by discarding these unnecessary corrections. This way of "doctoring numbers" is not a real solution (not an another theory), but a "stopgap", and guessing better QED formulations is in order.
So, the difficulties arise due to extrapolation of some theories beyond their domain of applicability and because of our human inperfections in guessing better theories.
A: Singularities or aporia in physics are avoided for the same reason why they are elsewhere: they are not easy to deal with, and may require the invention of new concepts of physics.
On the other hand, for that small band of  physicists who are interested in coming up with new concepts of physics, they are as nectar to a bee. 
