×) Counterexample: Consider the manifold M := R³\B, where B is the closed unit ball, equipped with the standard Euclidean metric. This manifold is certainly not Cauchy-complete and we can reach the singularity at r=1 in finite time. Now, if we had to define the dimension of the singularity, what dimension n should it have? n=2 (a sphere)? Maybe. At least we could extend M by the unit sphere to make it complete. But could the singularity also be a point (i.e. n=1)? Yes, certainly. By diffeomorphism invariance, we could simply find new coordinates and map R³\B to R³\{0}, so the singularity would suddenly become a point. So, as you can see, interpreting the singularity as a point or set of points that have a topological dimension doesn't work.

>A black hole itself (the singularity) is 1 dimensional - a single infinitesimal point.

In Euclidean geometry,

A cube is 3 dimensions.

A plane is 2 dimensions.

A line is 1 dimension.

A point is ...

> In this diagram the singularity is a line in spacetime i.e. a one dimensional object in spacetime.

is wrong or at least very misleading – the answer does (correctly) say that asking for the "dimensionality of a singularity […] is a meaningless question because the spacetime geometry is undefined at a singularity".