Singularities/ringularities and their properties
You need to be a bit cautious about statements like: A black hole contains a singularity at its center. It is a zero-dimensional point, and it's where all its mass is located. What you are referring to is a mathematical structure called the Schwarzschild metric. This is indeed a solution to the Einstein equations for a static black hole, but this particular solution takes an infinite time to form. So given that the universe is only a shade under 14 billion years old we can be confident that as I type this there are no Schwarzschild black holes in the universe.

Indeed, it is far from clear that a Schwarzschild black hole could ever form, even given infinite time, because black holes evaporate. This point is pursed in the question Why does Stephen Hawking say black holes don't exist?

Incidentally, the Schwarzschild black hole doesn't have all its mass at the centre. In fact it is a vacuum solution and contains no matter at all. The mass of the Schwarzschild black hole is actually a geometrical property called the ADM mass.

We run into the same problems with the spinning black hole. The structure with the ring singularity is the Kerr metric, and again while it's a perfectly valid solution to Einstein's equations, like the Schwarzschild metric it also takes an infinite time to form. Actually the Kerr metric is more troublesome still, because it is uncertain whether the structure is stable with respect to perturbations. It seems very likely that real collapsing stars would not form a structure resembling the Kerr metric. More precisely the region outside the event horizon would be well described by a Kerr metric, the region near the singularity would be different.

But if we put these doubts aside and return to what you asked, the answer is that in the Kerr metric the ring singularity really is a ring and not a collection of points.