General relativity expressed in terms of differential geometry. And it lets you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the equirectangular projection has a whole line at the top and bottom that correspond to North and South poles; or a single coordinate may refer to multiple points, for example by using inversive geometry the origin refers to all the points infinitely far away.

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?

General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the equirectangular projection has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.

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?