Do singularities have a "real" as opposed to mathematical or idealized existence? I was thinking of, for example a Schwarzchild metric at r=0, i.e. the gravitational singularity, a point of infinite density. I realise that there are different types of singularities--timelike, spacelike, co-ordinate singularities etc. In a short discussion with Lubos, I was a bit surprised when I assumed they are idealized and I believe he feels they exist. I am not a string theorist, so am not familiar with how singularities are dealt with in it.  In GR, I know the Penrose-Hawking singularity theorems, but I also know that Hawking has introduced his no-boundary, imaginary time model for the Big Bang, eliminating the need for that singularity. Are cosmic strings and other topological defects singularities or approximations of them (if they exist).  In what sense does a singularity exist in our universe? --as a real entity, as a mathematical or asymptotic idealization, as a pathology in equations to be renormalized or otherwise ignored, as not real as in LQG, or as real in Max Tegmark's over-the-top "all mathematical structures are real"? 
 A: Dear Gordon, I hope that other QG people will write their answers, but let me write mine, anyway.
Indeed, you need to distinguish the types of singularities because their character and fate is very different, depending on the type. You rightfully mentioned timelike, spacelike, and coordinate singularities. I will divide the text accordingly.
Coordinate singularities
Coordinate singularities depend on the choice of coordinates and they go away if one uses more well-behaved coordinates. So for example, there seems to be a singularity on the event horizon in the Schwarzschild coordinates - because $g_{00}$ goes to zero, and so on. However, this singularity is fake. It's just the artifact of using coordinates that differ from the "natural ones" - where the solution is smooth - by a singular coordinate transformation.
As long as the diffeomorphism symmetry is preserved, one is always allowed to perform any coordinate transformation. For a singular one, any configuration may start to look singular. This was case in classical general relativity and it is the case for any theory that respects the symmetry structure of general relativity.
The conclusion is that coordinate singularities can never go away. One is always free to choose or end up with coordinate systems where these fake singularities appear. And some of these coordinate systems are useful - and will remain useful: for example, the Schwarzschild coordinates are great because they make it manifest that the black hole solution is static. Physics will never stop using such singularities. What about the other types of the singularities?
Spacelike singularities
Most famously, these include the singularity inside the Schwarzschild black hole and the initial Big Bang singularity.
Despite lots of efforts by quantum cosmologists (meaning string theorists working on cosmology), especially since 1999 or so, the spacelike singularities remain badly understood. It's mainly because they inevitably break all supersymmetry. The existence of supersymmetry implies the existence of time-translational symmetry - generated by a Hamiltonian, the anticommutator of two supercharges. However, this symmetry is brutally broken by a spacelike singularity.
So physics as of 2011 doesn't really know what's happening near the very singular center of the Schwarzschild black hole; and near the initial Big Bang singularity. We don't even know whether these questions may be sharply defined - and many people guess that the answer is No. The latter problem - the initial Big Bang singularity - is almost certainly linked to the important topics of the vacuum selection. The eternal inflation answers that nothing special is happening near the initial point. A new Universe may emerge out of its parent; one should quickly skip the initial point because nothing interesting is going on at this singular place, and try to evolve the Universe. The inflationary era will make the initial conditions "largely" irrelevant, anyway. However, no well-defined framework to calculate in what state (the probabilities...) the new Universe is created is available at this moment.
You mentioned the no-boundary initial conditions. I am a big fan of it but it is not a part of the mainstream description of the initial singularity as of 2011 - which is eternal inflation. In eternal inflation, the initial point is indeed as singular as it can get - surely the curvatures can get Planckian and maybe arbitrarily higher - however, it's believed by the eternal inflationary cosmologists that the Universe cannot really start at this point, so they think it's incorrect to imagine that the boundary conditions are smooth near this point in any sense, especially in the Hartle-Hawking sense.
The Schwarzschild singularity is different - because it is the "final" spacelike singularity, not an initial condition - and it's why no one has been talking about smooth boundary conditions over there. Well, there's a paper about the "black hole final state" but even this paper has to assume that the final state is extremely convoluted, otherwise one would macroscopically violate the predictions of general relativity and the arrow of time near the singularity.
While the spacelike singularities remain badly understood, there exists no solid evidence that they are completely avoided in Nature. What quantum gravity really has to do is to preserve the consistency and predictivity of the physical theory. But it is not true that a "visible" suppression of the singularities is the only possible way to do so - even though this is what people used to believe in the naive times (and people unfamiliar with theoretical physics of the last 20 years still believe so).
Timelike singularities
The timelike singularities are the best understood ones because they may be viewed as "classical static objects" and many of them are compatible with supersymmetry which allowed the physicists to study them very accurately, using the protection that supersymmetry offers.
And again, it's true that most of them, at least in the limit of unbroken supersymmetry and from the viewpoint of various probes, remained very real. The most accurate description of their geometry is singular - the spacetime fails to be a manifold, i.e. diffeomorphic to an open set near these singularities. However, this fact doesn't lead to any loss of predictivity or any inconsistency.
The simplest examples are orbifold singularities. Locally, the space looks like $R^d/\Gamma$ where $\Gamma$ is a discrete group. It's clear by now that such loci in spacetime are not only allowed in string theory but they're omnipresent and very important in the scheme of things. The very "vacuum configuration" typically makes spacetime literally equal to the $R^d/\Gamma$ (locally) and there are no corrections to the shape, not even close to the orbifold point. Again, this fact leads to no physical problems, divergences, or inconsistencies.
Some of the string vacua compactified on spaces with orbifold singularities are equivalent - dual - to other string/M-theory vacua on smooth manifolds. For example, type IIA string theory or M-theory on a singular K3 manifold is equivalent to heterotic strings on tori with Wilson lines added. The latter is non-singular throughout the moduli space - and this fact proves that the K3 compactifications are also non-singular from a physics viewpoint - they're equivalent to another well-defined theory - even at places of the moduli spaces where the spacetime becomes geometrically singular.
The same discussion applies to the conifold singularities; in fact, orbifold points are a simple special example of cones. Conifolds are singular manifolds that include points whose vicinity is geometrically a cone, usually something like a cone whose base is $S^2\times S^3$. Many components of the Riemann curvature tensor diverge. Nevertheless, physics near this point on the moduli space that exhibits a singular spacetime manifold - and physics near the singularity on the "manifold" itself - remains totally well-defined.
This fact is most strikingly seen using mirror symmetry. Mirror symmetry transforms one Calabi-Yau manifold into another. Type IIA string theory on the first is equivalent to type IIB string theory on the second. One of them may have a conifold singularity but the other one is smooth. The two vacua are totally equivalent, proving that there is absolutely nothing physically wrong about the geometrically singular compactification. We may be living on one. The equivalence of the singular compactifications and non-singular compactifications may be interpreted as a generalized type of a "coordinate singularity" except that we have to use new coordinates on the whole "configuration space" of the physical theory (those related by the duality) and not just new spacetime coordinates.
It's very clear by now that some singularities will certainly stay with us and that the old notion that all singularities have to be "disappeared" from physics was just naive and wrong. Singularities as a concept will survive and singular points at various moduli spaces of possibilities will remain there and will remain important. Physics has many ways to keep itself consistent than to ban all points that look singular. That's surely one of the lessons physics has learned in the duality revolution started in the mid 1990s. Whenever physics near/of a singularity is understood, we may interpret the singularity type as a generalization of the coordinate singularities.
At this point, one should discuss lots of exciting physics that was found near singularities - especially new massless particles and extended objects (that help to make singularities innocent while preserving their singular geometry) or world sheet instantons wrapped on singularities (that usually modify them and make them smooth). All these insights - that are cute and very important - contradict the belief that there's no "valid physics near singularities because singularities don't exist". Spacetime manifolds with singularities do exist in the configuration space of quantum gravity, they are important, and they lead to new, interesting, and internally consistent phenomena and alternative dual descriptions of other compactifications that may be geometrically non-singular.
A: The singularity in a Schwarzschild metric is a three dimensional space where the Weyl curvature diverges.  There are reasons why we might think this is “quantized,” for otherwise the quantum evaporation would present the outside world with a singularity.  
An observer that follows a string into a black hole will observe the tidal forces on the string increase.  It is extended along the radial direction and its energy increases.  The extension is along the $X^+$ direction, and the gauge for the system is set on the $X^-$ direction.  This is opposite the gauge condition the exterior observer employs in observing the string fill up the stretched horizon of the black hole.  The analysis for the string on the horizon can be found in Susskind & Lindsey “Black Holes, Information and the String Revolution”  If it is desired I can work out the case of the string which falls into the black hole.  
The string falling towards the singularity becomes highly excited and reaches the singularity in some state.  It probably ends as an open string on a brane dual to the NS5-brane.  The string ends in this state of affairs, or it reaches its upper temperature limit and becomes something we are not well acquainted with.  
The density of states for a string with respect to modes n is
$$
\eta(n)~\sim~ exp(4\pi n \sqrt{\alpha’})
$$
that defines a partition function $Z~=~ \int \eta(n)exp(-n/T)dn$.  The temperature is computed by $1/T~=~\partial Z/\partial n$ and the path integral diverges for a temperature greater than 
$$
T_H~=~4\pi \sqrt{\alpha’}
$$
which is the Hagedorn temperature.  This is proportional to the reciprocal of the string length. The entropy of the system is the logarithm of the density of states the $S~\sim~ 1/nT_H$, which in the large n limit is zero.  The modes number is given by $n~=~1/(\sqrt{d}M_s)$, for d the number of degrees of freedom and $M_s$ the string mass.  
These physical conditions occur before the Planck energy is reached.  Consequently the Schwarzschild singularity is valenced by quantum mechanics.  The singularity is valenced by, or really replaced with, a D-brane with a gas of II strings or a D-brane with a gas of D0-brane solitons or … .
A: Of course, not. Singularities are prohibited by the Cosmic Censorship principle.
Even more: no matter can ever move under the so-called "event horizon" because as one approaches the horizon or any other potential well, the time slows, and on the horizon it slows infinitely. Thus if the horizon existed, nobody could reach it in finite time, and the black hole existence time is finite, so any black hole would evaporate before the falling observer reaches the event horizon. 
The singularity and event horizon terminology is useful only in theory which does not account for quantum effects and thermodynamics.
