A black hole is defined by its event horizon which starts from what is called a crease set, which is a set of events*, which are isolated in the sense that no messages can be sent between any of the events in the set.

A heuristic way of thinking about it is that a bunch of "mini-black holes" will form instantaneously at a set of points that are isolated from each other, each "mini black hole" expanding outwards from that point before merging together. In relativity whether two isolated events happen at the same time is subjective, so it is a moot question as to whether they form at the same time. For a collapsing star, effectively the merging happens very quickly and in the very early lifetime of the black hole.

Though this describes the formation of the event horizon, it is not correct to think of the singularity also forming at the crease set. All we can say about the singularity really is it is contained within the event horizon.

*An event is a point in spacetime, i.e. something with an exact location in both space and time.

https://www.worldscientific.com/doi/10.1142/S0218271811019220