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.
If you're not up to speed with general relativity this is going to be hard to explain, but I'll give it a go. The more determined reader may want to look at this PDF (just under 1MB in size) that describes the collapse in a rigorous way.
A couple of points to make before we start: you're being vague about the distinction between the singularity and the event horizon. The singularity is the point at the centre of the black hole where the curvature becomes infinite. The event horizon is the spherical surface that marks the radial distance below which light cannot escape. As you'll see, these form at different times.
The other point is that to make the calculation possible at all we have to use a simplified model. Specifically we assume the collapsing body is homogeneous (actually I see you anticipated that in your answer) and is made up of dust. In general relativity the term dust has a specific meaning - it means matter that is non-interacting (apart from gravity) and has zero pressure. This is obviously very different from the plasma found in real stars.
With the above simplifications the collapse is described by the Oppenheimer-Snyder model, and it turns out that the size of the collapsing object is described by the same equation that describes the collapse of a closed universe. This equation is called the FLRW metric, and it gives a function called the scale factor, $a(t)$, that describes the size of the ball of dust. For a closed universe the scale factor looks something like:
(image from this PDF)
A closed universe starts with a Big Bang, expands to a maximum size then recollapses in a Big Crunch. It's the recollapse, i.e. the right hand side of the graph above, that describes the collapse of the ball of dust.
The radius of the ball is proportional to $a(t)$, so the radius falls in the same way as $a(t)$ does, and the singularity forms when $a(t) = 0$ i.e. when all the matter in the ball has collapsed to zero size.
As always in GR, we need to be very careful to define what we mean by time. In the graph above the time plotted on the horizontal axis is comoving or proper time. This is the time measured by an observer inside the ball and stationary with respect to the grains of dust around them. It is not the same as the time measured by an observer outside the ball, as we'll see in a bit.
Finally, we should note that the singularity forms at the same time for every comoving observer inside the ball of dust. This is because the ball shrinks in a homogeneous way so the density is the same everywhere inside the ball. The singularity forms when the density rises to infinity (i.e. the ball radius goes to zero), and this happens everywhere inside the ball at the same time.
OK, that describes the formation of the singularity, but what about the event horizon. To find the event horizon we look at the behaviour of outgoing light rays as a function of distance from the centre of the ball. The details are somewhat technical, but when we find a radius inside which the light can't escape that's the position of the event horizon. The details are described in the paper by Luciano Rezzolla that I linked above, and glossing over the gory details the result is:
This shows time on the vertical axis (Once again this is comoving/proper time as discussed above) and the radius of the ball of dust on the horizontal axis. So as time passes we move upwards on the graph and the radius decreases.
It's obviously harder for light to escape from the centre of the ball than from the surface, so the event horizon forms initially at the centre of the ball then it expands outwards and reaches the surface when the radius of the ball has decreased to:
$$ r = \frac{2GM}{c^2} $$
This distance is called the Schwarzschild radius and it's the event horizon radius for a stationary black hole of mass $M$. So at this moment the ball of dust now looks like a black hole and we can no longer see what's inside it.
However note that when the event horizon reaches the Schwarzschild radius the collapse hasn't finished and the singularity hasn't formed. It takes a bit longer for the ball to finish contracting and the singularity to form. The singularity only forms when the red line meets the vertical axis.
Finally, one last note on time.
Throughtout all the above the time I've used is proper time, $\tau$, but you and I watching the collapse from outside measure Schwarzschild coordinate time, $t$, and the two are not the same. In particular our time $t$ goes to infinity as the ball radius approaches the Schwarzschild radius $r = 2GM/c^2$. For us the part of the diagram above this point simply doesn't exist because it lies at times greater than infinity. So we never actually see the event horizon form. I won't go into this any further here because it's been discussed to death in previous questions on this site. However you might be interested to note this is one of the reasons for Stephen Hawking's claim that event horizons never form.
Best Answer
The Schwarzschild metric for $d$ dimensions is the standard form $$ ds^2~=~-e^{2\phi}dt^2~+~e^{2\gamma}dr^2~+~r^2d\Omega^2 $$ These metric terms in the Einstein field equation gives $$ R_{tt}~-~\frac{1}{2}Rg_{tt}~=~G_{tt}~=~ -e^{2\phi}\Big((d~-~1)\frac{e^{2\gamma}}{r} ~+~\frac{(d~+~1)(d~-~2)}{2r^2}(1~-~e^{2\gamma}\Big) $$ A multiplication by $g^{tt}$ removes the $-e^{2\phi}$ and we equate this with a pressureless fluid $T^{tt}~=~\kappa\rho$. So we think of the black hole as composed of “dust.” Some analysis on this is used to compute the $G_r^r$ gives the curvature term $$ G_r^r~=~\Big((d~-~1)(\phi_{,r}~+~\gamma_{,r})\frac{e^{-2\gamma}}{r}~+~\kappa\rho\Big). $$ which tells us $\phi~=~-\gamma$, commensurate with the standard Schwarzschild result, and that $$ e^{-2\gamma}~=~e^{2\phi}~-~\Big(\frac{r_0}{r}\Big)^{d~-~2}. $$ The entropy of the black hole is then computed by writing the density according to these metric elements and computing the Rindler time coordinates $S~=~2\pi(d~-~2)A/\kappa$.
The results more or less follow as with the standard $3~+~1$ spacetime result. The Connection with strings is to work with the entropy of the black hole. The $1~+~1$ string world sheet has $d~-~1$ transverse degrees of freedom which contain the field data. The entropy $S~=~2\pi(d~-~1)T$ may be computed with the string length, which reduces to the holographic results in $d~=~4$ spacetime.
The event horizon is $d~-~2$ dimensional, which for $10$ dimension means the horizon is $8$ dimensional. The singularity is not considered in these calculations. The factors $e^{-2\gamma}~=~e^{2\phi}$ become extremely large. The metric approximates $$ ds^2~\simeq~\Big(\frac{r_0}{r}\Big)^{d~-~2}dt^2~+~r^2d\Omega^2 $$ which is a $d~-~1$ dimensional surface where the Weyl curvature diverges for $r~\rightarrow~0$. For $d~=~4$ this has properties similar to an anti-deSitter space.
The theory of black holes essentially follows in arbitrary dimensions. It is interesting to speculate on what the singularity is from a stringy perspective. The event horizon contains the quantum field information which composes the black hole. This may then have some type of correspondence with the interior singularity, with one dimension larger. For a black hole that is very small $\sim~10^3$ Planck units, the horizon is a quantum fluctuating region, as is the singularity, and the QFT data on the two may have some form of equivalency.