This write-up will be primarily concerned with the concept of a space-time singularity, for example the one that lies at the heart of a black hole. As noted above, the singularity is a region of space and time where the current accepted physical models; relativity and the quantum mechanical standard model, can make no predictions and offer no insight to what lies within that region.

When an object forms a black hole its gravity overwhelms its matter, crushing it into a smaller and smaller region, and physics as yet provides no mechanism for this collapse to stop. Logically the matter occupies a smaller and smaller region of space-time, all the way down to infinitely small. In 1965 Roger Penrose in fact proved that singularities must occur in gravitational collapses, regardless of the symmetry or other properties of the initial mass.
Right now physics has enough tools to model this collapse right down to the Planck length.When you use a microscope to observe something small, you are limited by the wavelength you are using. A light microscope can only resolve details the size of the wavelength of visible light. Due to the wave particle duality nature of quantum mechanics however, you can use particles such as electrons to resolve smaller details. If you boost these particles to higher energies, their frequency increases, and you can again resolve smaller details. In a way that's what particle accelerators are for; particles of such high energy are used, you can resolve the very fundamental particles that make up everything else. When you get down to regions the size of the Planck length however, the wavelength you need has an mass/energy (E=mc^2) sufficiently large enough to form a black hole! However, if there is (as relativity and the standard model suggest), no limit to the 'smallness' of space, then there is still (in the same way you have an infinite number of integers, and an infinite number of reals between zero and one), the planck volume has still has 'space' for an infinity of things to happen!

The next physical models (such as super-string theory, loop quantum gravity) do not have this problem, as space/time is quantised; there is a fundamental unit to everything, past which or course you can't see. In fact there's hints that the 'theory of everything' must be 'background independent', that is the physics doesn't take place on some abstract mathmatical background, called space/time, rather space/time is a patchwork of discreet entities. As you've defined what the smallest bit of space is, then this must be a singularity in this formulation.

Do Singularities really matter anyway?

As I said physics can offer no answers as to what lies within the (possibly) infinitely large region from the Planck size to the singularity. For a long time physicists hoped the question was irrelevant as it appears anything massive enough to collapse its matter down to infinity will be a black hole and therefore have an event horizon associated with it. Originally this event horizon was a one-way membrane, you can put matter, energy and information past it, but nothing can come back out of it. Any singularity hidden inside the black hole can therefore never affect the rest of the universe, ever, and can therefore be forgotten about. This cosmic censorship hypothesis (suggested by Penrose in 1969) says you can never have a naked singularity; it must always be clothed by an event horizon.

Sleeping dogs have a habit of waking up....

Of course really you can't just let the problem lie there, several important hypothesis that stem from the accepted correct (if incomplete) physics mean you have to seriously think about the consequences of allowing physics to 'make' a singularity.
Firstly cosmology has long sought to explain the origin and evolution of the universe. Observations by Edwin Hubble seemed to suggest all the galaxies are moving away from each other, at a rate proportional to their distance from us. This implied that once they were very close together, in fact tracing backwards infinitely close together...This lead to the formulation of the big bang theory, where the entire universe essentially exploded from an infinitely small region; of course this is a singularity!. Every observation made has so far has confirmed some kind of big bang occurred, refinements such as cosmological inflation don't alter the fundamental fact that the theory must have contain a singularity, a fact proved by Hawking. As you can't see past the Planck length, you can't make predictions what came out of the singularity at the dawn of time, in fact as I said above you could regard the evolution of the universe from the singularity up past the Planck length to have as rich a history as our own universe since the Planck length. As what came before must determine what comes after, cosmology has a real problem with singularities..... Secondly work by Professor Stephen Hawking and other have shown black holes are not in fact completely black, and do in fact radiate at a wavelength proportional to their size. (Please see Hawking radiation for more). A consequence of this is they might radiate away, (over time) all their energy, which could leave a naked singularity behind. What effect a naked singularity would have on the rest of the universe, I don't think anybody knows, I'm pretty sure you can't in fact calculate the effect of this space-time infinity.

Also it's just occurred to me, if you allow sizes smaller than the Planck length to exist, (even if you can't measure them) then when a black hole decays past a certain point, it can emit radiation/particles of sufficient energy to again be black hole, containing a singularity. This would be a self-perpetuating growth, a free lunch of infinite size, something, which I personally do not believe, is possible.

Out with the old in with the new?

So the 'old' physics seems to predict singularities as a logical consequence, but cannot offer any theories of their behaviour; the mathematics simply breaks down. The current hot 'new' physics is superstring theory and its partner m-theory. In these space-time is quantised in that it can only come in 'packets' limited to about the Planck length in size. These strings or branes do away with the concept of infinitely small and in doing solve a lot of problems in physics. In these theories (and there are many, and no-one knows how the one that describes our universe came to be chosen) a black hole would collapse to a string or a brane and no further. The question then arises can the string/brane that is the end product of the gravitational collapse a.k.a. the singularity, contain the necessary energy and information necessary to describe the black hole?

My own two-penneth

I think in this string/brane picture the cosmic censorship can be maintained, I believe that the singularity becomes a topologically complicated knot of 11 dimensional space-time. The emission of particles from the event horizon represents decay of this 'singular' knot, as it decays, it loses energy/mass and the horizon shrinks. At the point where the horizon shrinks to nothing, the 'singularity' finally decays in a flash of Hawking radiation. I think by redefining the singularity in such a way might help some of the problems involved in black hole entropy also, (I humbly refer to my node there...).
Recent work in knot theory has shown that knots may in fact be quantised also. This would mean that some modes of decay for knot/singularity might well be forbidden, which could give rise to 'absorbance lines' in the spectra of black hole radiation. If two cosmic rays of sufficient energy were to collide they could form a black hole in the order of the Planck size, and the above effect could be seen as it decays....

This w/u was done in response to the The content rescue team:nodes Largely of the top of my head, any errors/typos/glaring omissions please msg me!