Why are we scared of singularities? I often hear people say that general relativity predicted its own demise because of the singularities it predicts. If I'm not mistaken, this is also a problem in QFT. I wonder why singularities garner so much hate. To me, it seems like negative or complex numbers: we used to hate these things but now they are more generally accepted. Why can't it be that nature just has singularities? We might not get it, we might not like it, but it may be so.
 A: Those infinities can mean the breakdown of the theory. Blackbody radiation predictions using classical theory says that a blackbody will radiate an infinite amount of energy. It meant the theory breaks down, and it was only able to be explained right with quantum theory.
For General Relativity (GR) it is the same situation. At true singularities (not coordinate singularities which transfer away under a change of coordinates, like the black hole horizons, which are not the same problem, though since they tend to hide singularities maybe also means something useful for quantizing gravity) some of the scalar curvature values will become infinite, and since scalar this can not be fixed with a coordinate transformation. We need a quantum gravity theory to explain the black hole singularities. Or something. 
But in GR it is not simply that, it also means that the spacetime where that singularity exists is incomplete - at the singularity the geodesics simply end. If you are traveling on one, and manage to still be something as you near the singularity, in very short and finite proper time your worldline will end. Like they tend to do in your palm, but this would be real, if it existed. Well, that's not so bad because nobody is stupid enough to go there. 
But it has other physical implications on that spacetime. It also says that the spacetime is not predictable, that causality is violated there. And if geodesics a-causally end at the singularity, anything could also come out of the singularity, and extend over that spacetime. It would just not be causally safe. A dragon could come out or your 10,000 years old ancestors riding a rocket ship. That spacetime becomes acausal. 
It is causally disconnected from the spacetime outside the black hole due to the horizon, so none of that makes any difference outside. But if there was a black hole without a horizon, then all of the connected spacetime would be acausal. Such a singularity is called a naked singularity. There is a censorship principle that says that there are no naked singularities. It's a conjecture, not proven mathematically, once in a while somebody finds a possible but pretty strange and typically considered not physically plausible solution of the GR equations, with a naked singularity. 
I still agree with @CuriousMind, real singularities probably don't exist. But it does mean that the currently best known theories cannot be all right, that there needs to be a solution to the singularity problems, probably with a still unknown quantum gravity theory. 
Hope you now don't fret too much if you might not be living in a causal spacetime. But if it drives you to find the solution, good for physics. 
A: 
To me it seems like negative or complex numbers.

Not at all. Negative or complex numbers are useful tools that, while maybe not having a direct equivalence in physical terms (you cannot have "-3 atoms" in a volume and no ruler will give you complex numbers as a readout), they are just a mechanism which collapse down to "realistic" numbers at the end, if you are so inclined to actually perform a real world calculation.
Note that not every infinity is a singularity. Some infinities have their uses; be it in infinitesimals (=> differentials, integrals), real/imaginary/transcedental etc. numbers (no matter whether they physically exist in the real world), or thinking about what happens if you keep going and going and going... those are just mathematical tools, based on the general concept of infinity, to get the job done. 
Not so a singularity. 1/0 can never, by any means, be undone by further calculations - it's over. Undefined. Wall of bricks across the street. Yes, you can get close by shrinking some Epsilon, but that's not what people are talking about. They are interested in what is actually physically there. Never being able to reach that spot which is clearly encapsulated by an Epsilon-volume is not fun.
This is all very much in layman's terms. But it might show why people view singularities differently than those other phenomena.
A: 
"To me it seems like negative or complex numbers. We used to hate these things but now they are more generally accepted. "

Indeed. And in a general context, the infinite answer that some equations return might not be a problem. We have all kinds of rigorous notions of infinite quantities in set theory; see, for example, aleph and beth numbers and infinite ordinals) and topology: for example compactification of sets through the addition of a point at infinity. In the context of the Riemann sphere, the $z_\infty$ returned by a meromorphic function at a pole is altogether reasonable - it's a point on the sphere utterly like any other. 
Crucially, though, in physics, most numerical results of theories either correspond to measurements, potential measurements or at least influence potential measurements in such a way that singularities imply infinite readings for potential measurements. Most physicists would agree that an infinite voltage is not a realistic measurement from a voltmeter. 
Physics is not mathematics: it is mathematics together with the need to make descriptions wrought in mathematical language correspond to real world, experimental observation. Since no one has ever witnessed an infinite reading (unless you do something really daft and decide to define a voltage scale that is the tangent of an SI voltage reading in volts, for example), we reason inductively (not mathematically inductively) that no reasonable measurement will ever be infinite.
A: Science is based on observation, and every event horizon shows that observation is restricted by some screens, and that we will never be able to observe what is found behind the screen (except if we are travelling ourselves behind the event horizon, with awkward consequences).
The lack of tolerance of singularities by science is also due to a lack of correct understanding of the principle of relativity: The twin paradox shows that observation and reality are two things, observation does not correspond always to reality: The twin having made a round trip near speed of light is observed having returned after 20 years, but in reality he aged only by 5 years. It is exactly the same principle which makes us observe infinities near event horizons. The twin paradox is a case of special relativity, and the event horizon is a case of the Schwarzschild metric.
If we measure and observe infinite values, that does not mean that there  i s  infinity, and general relativity does not "break down" when some event horizon prohibits the observation of certain regions of the universe.
Spacetime is relative, and our spacetime is our manifold of observation. If our observation is not able to follow certain geodesics that does not mean that this geodesic is not following the laws of GR (where it is not excluded that the laws of GR are completed by some other laws, quantum theoretical or not).
A: In some sense infinities are an integral part of renormalisable field theories. There measurable quantities can be computed naively and turn out to be infinite. This can however not be right because no measuring apparatus goes up to infinity. Put another way, particles propagate, they must have a finite mass. Finiteness of measurements is a (somewhat tautological) experimental fact. Field theories are therefore 'fixed' by absorbing the infinities in quantities that are not observable.
In a Renormalisation Group (RG) language, at every (spatial) scale there is a different effective theory. The theory at the observation scale contains (by definition) finite parameters. When RG flow equations are solved and smaller scales are probed, it turns out that some parameters grow without bound. Therefore, if space is continuous, then the truly microscopic parameters are infinite. If you insist on defining a mass for the electron at the smallest possible scale, it must be infinite. We can't really measure it, but we can measure how it grows as the observation scale $s$, is lowered,
$$m(s) = \frac{m_0}{s^{\eta}} \, .$$
$\eta$ is a positive exponent. $m_0$ is finite and measurable. $m(0)=\infty$ would be the microscopic mass.
I think that we don't run away from infinities. It is just that by definition they are not measurable. Finite numbers can be compared to each other. They are therefore measurable. Infinities on the other side are all the same. Sometimes they can be converted to finite numbers with constructions such as the RG or the Riemann sphere and related to something physical.
A: It is entirely consistent with the mathematics of general relativity that the universe does not exist at or beyond a 20,000 kilometer sphere surrounding the Earth. Since there is no universe at that boundary, GR can make no predictions about what happens there, and thus allows any arbitrary thing to happen there.
Obviously this is unsatisfying, but it also has a trivial fix: the mathematics of GR can be continuously extended to and beyond 20,000 sphere, so we simply insist that it does so. We have no reason to believe otherwise, anyways.
There is (likely) a boundary infinitely far future and another infinitely far and past that would provide an obstacle to this continuation process, but that's okay because it's infinitely far in the future or the past, so isn't really relevant to anything.
The problem with the type of singularity that people make a big deal of in the context of GR is that it is forms an obstacle that can be reached in finite time.
