We say the big bang (initial singularity) didn't happen at a point, but is it the same with the singularity of a black hole? I have read this question:

The simple answer is that no, the Big Bang did not happen at a point. Instead, it happened everywhere in the universe at the same time. Consequences of this include



*

*The universe doesn't have a centre: the Big Bang didn't happen at a point so there is no central point in the universe that it is expanding from.




*

*The universe isn't expanding into anything: because the universe isn't expanding like a ball of fire, there is no space outside the universe that it is expanding into.


Did the Big Bang happen at a point?
Now this has been puzzling me for a while, does the singularity of a black hole happen at a point, or does it happen everywhere at the same time:

*

*The black hole does not have a centre: the singularity does not happen at a point, so there is no central point in the black hole


*the black hole isn't squeezing into anything: because the black hole isn't squeezing like a ball of fire, there is no space that it would be squeezing into
Now on this site, many draw similarities between the singularity of the big bang, and the singularity of a black hole, in that the initial singularity (big bang) was the start from where space expanded, and in the case of a black hole space contracts into a singularity. Now if the big bang did not happen at a point, but happened everywhere at the same time, then does the contraction of space in a black hole happen (does the singularity "happen") everywhere at the same time?
Question:

*

*We say the big bang (initial singularity) didn't happen at a point, but is it the same with the singularity of a black hole?

 A: Not exactly. In this answer, I'll deal mainly with Schwarzschild black holes, which are a bit simpler. I'll comment on more general black holes at the end, but I'll leave for someone else to cover the details.
Schwarzschild Black Holes
Both the Big Bang and black hole singularities (reminder: I'm assuming a Schwarzschild black hole for now) are spacelike singularities. This means essentially that they do not happen at a place, but at a time. However, they are still considerably different.
All observers arose from the Big Bang. Hence, in this sense, it did happen everywhere at once. You will never find an observer (i.e., a path through spacetime) which did not start at the Big Bang.
Black holes, on the other hand, are a bit different. You will find observers who will never fall into a black hole, and hence will never hit the singularity inside it. In this sense, the singularity is not everywhere, because not every observer will crash into it. It is possible, for example, for observers to go to infinity in infinite time.
Hence, it is correct to say the singularity at a black hole doesn't happen at a (spatial) point. It is never "to the right" of an observer, for example. It is to the future. However, there are also some differences when we compare it with the Big Bang, so, at least in my opinion, it is not fair to say the singularity happens everywhere (it happens inside the black hole).
Charged and Rotating Black Holes
For Reissner–Nordstrom (charged) and Kerr (rotating) black holes (or for the more general Kerr–Newman (charged and rotating) black holes), the singularity is not spacelike. In fact, you end up with a fair more complicated structure inside the black hole, and there is the possibility of an observer simply navigating through the black hole without ever hitting the singularity (this is impossible in Schwarzschild, in which every observer that enters the black hole hits the singularity in finite time). Hence, this situation is way more different from the Big Bang singularity, and the singularities do happen at a place, not an instant. PBS Spacetime discussed a bit of the Kerr structure in the episode Mapping the Multiverse, in case you're interested.
It is worth recalling that these solutions assume high levels of symmetry and might not reflect what actually happens after stellar collapse. In real situations, the singularities might be fairly different.
Edit: as pointed out by safesphere in the comments, spinning a Schwarzschild black hole will lead to the singularity becoming a ring-like structure. This time, it is unavoidable to cross a structure known as the Cauchy horizon, which is a null (lightlike) surface within the black hole. Since it is unavoidable for anything that fell inside the black hole, the Cauchy horizon happens everywhere within the Kerr black hole in the same sense the Schwarzschild singularity happens everywhere within the Schwarzschild black hole. A similar comment also holds for Reissner–Nordstrom black holes, although the singularity is not ring-like.
