Need for a coordinate-independent definition
multiple coordinates may refer to a single point
Normally, the way we define this kind of thing in GR is that we have an atlas, and the atlas is made of charts. Each chart is required to be invertible, so no, we can't have multiple coordinates that refer to a single point. In any case, when we define dimensionality on a topological space, we do it in a coordinate-independent way. E.g., one can use the Lebesgue covering dimension.
A singularity in the metric versus a singularity on a well-defined metric background
Suppose I have a two-dimensional space with coordinates $(u, v )$,
and I ask you whether $S = \{(u, v )|v = 0\}$ is a point or a curve,
while refusing to divulge what metric I have in mind. You’d probably say $S$ was a curve, and if the metric was $ds ^2 = du ^2 +dv ^2$, you'd be right. On the other hand, if the metric was $ds ^2 = v^2du ^2 +dv ^2$,
$S$ would be a point. This was an example where there were two possible metrics we could imagine. At a singularity, it’s even worse. There is no possible metric that we can extend to the singularity.
Hawking and Ellis have a nice discussion along these lines, including an example similar to the one above, in section 8.3, "The description of singularities," p. 276:
[The singularity theorems] prove the occurrence of singularities in a large class of solutions but give little information as to their nature. To investigate this in more detail, one would need to define what one
meant by the size, shape, location and so on of a singularity. This would
be fairly easy if the singular points were included in the space-time
manifold. However it would be impossible to determine the manifold
structure at such points by physical measurements. In fact there
would be many manifold structures which agreed for the non-singular
regions but which differed for the singular points.
After presenting the example, they say:
In the first case the singularity would be a three-surface, in the second case a single point.
Not a point or set of points
So is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate?
Well, technically it's none of the above. A singularity in GR is like a piece that has been cut out of the manifold. It's not a point or point-set at all. Because of this, formal treatments of singularities have to do a lot of nontrivial things to define stuff that would be trivial to define for a point set. For example, the formal definition of a timelike singularity is complicated, because it has to be written in terms of light-cones of nearby points.
Boundary constructions don't provide an answer
There are some possible heuristics you might use in order to describe the singularity as if it were a point-set and talk about its dimensionality as if it were a point-set. You can draw a Penrose diagram. On a Penrose diagram, a horizontal line represents a spacelike 3-surface, with one dimension being shown explicitly on the diagram and the other two being because rotational symmetry is implicit. If you look at the Penrose diagram for a Schwarzschild black hole, the singularity looks like a horizontal line. It's not a point-set, but if it were, it would clearly be spacelike, and it sort of looks like it would be a 3-surface. This is very different from what most people would probably imagine, which would be that it's a 1-dimensional timelike curve, like the world-line of an electron.
If you try to develop this heuristic into something more rigorous, it basically doesn't work. This program is referred to as "boundary constructions." Reviews are available on this topic (Ashley, Garcia-Parrado). There are a number of more or less specific techniques for constructing a boundary, with an alphabet soup of names including the g-boundary, c-boundary, b-boundary, and a-boundary. As someone who is not a specialist in this subfield, the impression I get is that this is an area of research that has turned out badly and has never produced any useful results, but work continues, and it is possible that at some point the smoke will clear. As a simple example of what one would like to get, but doesn’t get, from these studies, it would seem natural to ask how many dimensions there are in a Schwarzschild black hole singularity. Different answers come back from the different methods. For example, the b-boundary approach says that both black-hole and cosmological singularities are zero-dimensional points, while in the c-boundary method (which was designed to harmonize with Penrose diagrams) they are three-surfaces
(as one would imagine from the Penrose diagrams).
May depend on the type of black hole
People have studied GR in more than 3+1 dimensions, and, e.g., in 4+1 dimensions you can get things like "black rings." If you look at how these are actually described in the literature, people seem to find it more convenient to talk about the topology of the horizon, rather than the dimensionality of the singularity: http://relativity.livingreviews.org/Articles/lrr-2008-6/fulltext.html . This is presumably because the formalism is ill-suited to talking about the dimensionality of the singularity.
Another example is the Kerr metric for a rotating black hole. The singularity is commonly described as a ring. But in this review article, there is a discussion of the singularity on p. 8 and again on p. 28. In both spots, there are scare quotes around "ring." Again I think this is because we can't really answer geometrical questions about the singularity, because it isn't a point-set, and therefore you can't say what the metric looks like there.
A strong curvature singularity criterion doesn't provide an answer
Another approach is to look at what happens to matter that goes into the formation of a black hole through gravitational collapse, or to a hypothetical cloud of test particles that falls into an eternal black hole. If the matter is crushed to zero volume, then it might make sense to interpret this as evidence that the singularity should be thought of as having zero volume.
Unfortunately this doesn't necessarily give a definitive answer either. One can define something called a strong curvature singularity (s.c.s.), defined as one for which a geodesic is incomplete at affine parameter $\lambda=0$, with $\lim_{\lambda\rightarrow0}\lambda^2R_{ab}v^av^b\ne0$, where $v^a$ is the tangent vector. The volume of a cloud of test particles goes to zero as it approaches such a singularity, the interpretation being that infalling matter is crushed, not just spaghettified. A Schwarzschild spacetime's singularity is not an s.c.s., because it's a vacuum spacetime, so the Ricci tensor vanishes. That is, there is only spaghettification, not crushing. A cloud of test particles maintains an exactly constant volume as it falls in.
However, a completely different situation could possibly exist during the collapse leading to the formation of an astrophysical black hole. During the collapse, there is infalling matter present, so the Ricci tensor need not vanish. Indeed, it appears that in some fairly realistic models of gravitational collapse, the singularity, during the period of collapse, is a timelike (locally naked) singularity (Joshi), meaning that it is completely different in character from the spacelike singularity of an eternal black hole such as a Schwarzschild black hole. It appears that in such calculations, the density of matter does blow up at the singularity, which suggests that it may be an s.c.s. during formation.
Difficulties in the case of the Schwarzschild spacetime
When we think about a black hole, we usually imagine by default the kind of eternal black hole described by the Schwarzschild spacetime. Some significant difficulties occur even in this simplest possible case. As noted above, the singularity may have a completely different character than the singularity occurring during the gravitational collapse of an astrophysical black hole, and this leads to suspicion that by considering the Schwarzschild case, we are omitting essential considerations.
Furthermore, we have the no-hair theorem, which states that for a stationary electrovacuum spacetime having an event horizon, there is only one class of solutions, which can be parametrized by three variables: mass, charge, and angular momentum. This defines a clear sense in which the singularity of a stationary black hole has no physical properties. If it did have such properties, those would have to be limited to the list of three properties described by the no-hair theorem. However, these are not properties of the singularity but rather properties of some large region of spacetime as measured by a distant observer, who can't even say whether the singularity exists "now." (It may in fact appear to such an observer that infalling matter takes infinite time to pass through the horizon.)
The chances for a more definitive answer might be better in the case of a naked singularity. Such a singularity can exist in both the past light cone and the future light cone of an observer, so one can imagine doing experiments on it and finding the results. In this sense it may be more likely to have measurable properties.
References
Ashley, "Singularity theorems and the abstract boundary construction," https://digitalcollections.anu.edu.au/handle/1885/46055
Garcia-Parrado and Senovilla, "Causal structures and causal boundaries," http://arxiv.org/abs/gr-qc/0501069
Joshi and Malafarina, "All black holes in Lemaitre-Tolman-Bondi inhomogeneous dust collapse," https://arxiv.org/abs/1405.1146
