Do all black holes have a singularity? If a large star goes supernova, but not enough mass collapses to form a black hole, it often forms a neutron star. My understanding is that this is the densest object that can exist because of the Pauli exclusion principle: It's made entirely of degenerate matter, each particle of which cannot occupy the same quantum state of any other.
So these objects are so massive that they gravitationally lens light. If you make them more massive, they bend the light more. Keep going and going until they bend the light so much that light passing near the surface can barely escape. It's still a neutron star. Add a bit more mass, just enough that light passing just over the surface cannot escape. Now it's a black hole with an event horizon (I think?). Does this mean the neutron star has become a singularity? Isn't it still just a neutron star just beneath the event horizon?
Why are black holes treated as having a singularity instead of just an incredibly massive neutron star at its center? Does something happen when an event horizon is "created?"
 A: Wikipedia seems to indicate they they all do:

At the center of a black hole as
  described by general relativity lies a
  gravitational singularity, a region
  where the spacetime curvature becomes
  infinite. For a non-rotating black
  hole this region takes the shape of a
  single point and for a rotating black
  hole it is smeared out to form a ring
  singularity lying in the plane of
  rotation. In both cases the singular
  region has zero volume. It can also be
  shown that the singular region
  contains all the mass of the black
  hole solution. The singular region can
  thus be thought of as having infinite
  density.

A: If there is a trapped null surface, and one of the energy conditions like the null energy condition, or the weak energy condition is satisfied, and space outside is noncompact, there has to exist either a singularity or closed timelike curve inside the black hole.
See Penrose and Hawking.
A: So, you are hoping a neutron star's edge could be just under the event horizon and be stable there - in other words, the neutron star would not collapse into the singularity that would form.
That is not possible. See my answer here: Why can't you escape a black hole? It has a nice picture that explains that once you pass the event horizon, the curvature of space-time essentially rotates the time direction to point into a spatial direction towards the center of the black hole.  So just as you cannot resist moving forward in time, the edge of the neutron star cannot resist falling into the center of the black hole - that is it's future time direction.
A: This is a question for the Physics forum.
The honest answer is that we don't know for sure. General relativity is a classic (non-quantum) theory, so it should fall apart at very small scales and very high energy densities - exactly what's supposed to happen in a singularity. We're still waiting for the quantum relativity theory to emerge; if and when that happens, we'll know a lot more about singularities.
WRT black holes, it is perhaps prudent to say that, the deeper in you go, the less we actually know what's going on.
We know a lot about the region surrounding the event horizon; we're pretty confident we got that right, and actually we have observations nowadays matching the theory.
We believe we know a little about stuff happening within the event horizon, but things are getting a bit foggy there.
We can't in all honesty say that we know much about what happens down at the bottom, in the very singularity; that's where general relativity divides by zero and goes belly up.
So, take everything with a grain of salt and keep an open mind.
A: The maximum mass for a neutron star is the Tolman–Oppenheimer–Volkoff limit and is thought to be between 1.5 and 3 solar masses, the range being due to uncertainties of the equation of state of matter at these extreme densities. If the mass of a neutron star exceeds this limit then implosion to a black hole is assumed to be inevitable, there is no force that can repel the collapse according to general relativity.
So black holes are distinct from neutron stars and an event horizon only forms around the black hole.
The quantum degeneracy pressure of electrons that you mentioned occurs inside white dwarfs. In neutron stars the quantum degeneracy pressure of neutrons is responsible.
A: Short answer is yes.
But if you want to nit pick, I could argue that when a star collapses to form a BH, it first forms a horizon before the singularity forms (cannot form a "naked singularity"). And since time inside the horizon is essentially frozen with respect to that of an observer outside, the singularity NEVER forms. Yet from the point of view of the collapsing star, the singularity forms in about a millisecond after the horizon.
A: In classical General Relativity, once an event horizon forms, every particle inside that event horizon will inevitably travel in the direction of the center of the black hole. This is what is meant by "gravitational collapse" and how matter comes to form a singularity in the center- no matter how small it is, or how close to the center it is, nothing can prevent it from approaching ever closer to the center. From the point of view of the object itself, it does reach the center in a finite time.
In some more exotic theories of physics, such as string theory or loop quantum gravity, the quantized nature of space and time comes to the rescue and prevents a singularity from forming, so a maximum, finite density is reached and an equilibrium is maintained in the center. This is similar conceptually to what you describe, but still a more exotic and much, much denser material than neutron star-stuff.
The density we're talking about here would be approximately one Planck Mass per cubic Planck Length, in other words 2.176 51 × 10^−8 kg / (1.616 199 × 10^−35 m)^3 ~= 5.15556^96 kg/m^3, where neutron star material is "only" (roughly) 10^18 kg/m^3.
In either case, however, outside the event horizon, the black hole can be treated mathematically and observationally as a simple singularity, so for observational calculations, there is no "value added" in worrying about the inner workings of the black hole. The theorem describing this is colloquially called "Black holes have no hair." This theorem was proven and coined by John Wheeler, the same physicist who coined the phrase "black hole" in the first place.
A: All black holes contain singularities, however not all singularities involve black holes. A neutron star may be dense, matter the size of a pinhead can weigh as much as the earth, but there seems to be a mathematical cut-off point beyond which a black hole is formed. The first step for this is the formation of the even horizon, and everything within the event horizon is your singularity. If a neutron star's mass increases in relation to its radius to form its critical circumference (a star 10x heavier than our sun would have a critical circumference of around 110 miles, or a 20 mile radius), it undergoes gravitational collapse and you have your black hole. Beyond it, the matter is so infinitely dense that the gravitational pull sucks every photon of light into its centre. At this point you have your singularity, where the infinite density mean space and time as we know it, cease to exist. You find yourself in a constant state of chaotic equilibrium; like the ultimate drag car using up hundreds of kilos of fuel every second.
A: NO: all black-holes do NOT have a singularity.  The following description of a “radiation-structured” BH is conjectural;  but it answers your question about singularities, and it makes sense. I've omitted all but the key ideas in describing this concept, and there's a lot of detail that may, as yet, be unexplainable:
When particle momentum in a developing black-hole (not quite yet a BH) reaches a critical level, gravitational potential has produced a value of particle momentum that conflicts with the values allowed by quantum mechanics, and nature responds by transforming high-energy particle-dynamics into high energy radiation.  This radiation is manifested at or near the BH boundary, rather than in the form of a singularity.
This critical event establishes a BH boundary 'density' corresponding to E/surface-area = energy-density. This energy density remains constant regardless of BH size and energy content ; a formative-BH-star’s particle-driven temperature also remains constant as it’s size and energy and entropy increase, following the transformation of a proto- blackhole into a BH.  Additional energy, after BH formation, joins the BH boundary energy, rather than enters the BH interior. There is no singularity associated with this concept of a “radiation-structured” black-hole.  Comments are appreciated…RobertO
A: A black hole is a supper massive object with a very intense gravitational field, and so is a neutron star, but the difference between the two is, that, light can escape the gravitational field of a neutron star, but not that of a black hole, that is why it is called a black hole. 
Some stars are massive enough to become black holes, but some are not, they either become dwarf stars (like our sun would), or neutron stars. Stars above the chandrashekhar limit (about 2 solar masses)would become black holes, have an escape velocity more than the speed of light, while those below it wont. 
An event horizon is just light trying to escape a black hole, but stuck in an orbit around it. 
Singularity is just a hypothesis, nobody knows if it really exists or not, but its not necessary for a black hole (according to me). I believe in the exclusion principle, but that would be at the string level (string theory) at a scale of $10^-33$ metres and that's how far a 'singularity' can go.
