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"The singularity of a black hole is not a point in space it is not even a point in spacetime, in fact it is not even in spacetime. The singularity is not a point on the spacetime manifold. If it were it would obviously not be a singularity."

The source is unimportant.

I feel that a singularity is technically any location--in space-time--where properties, namely density, become infinite. And less technically, any location that we don't really feel can exist according to the understandings of generally excepted theory (i.e. infinite density again, information loss, blah blah blah).

Colloquially, a gravitational singularity has become synonymous with black-holes (perhaps unfortunately), in which case a black-hole (the juicy core in particular) very possibly occupies a finite, non-zero, region of space-time.

Is there any backing to "singularities" not occupying space-time?

If this is the case, how can they come into existence, and later(temporally) cease to exist?

How can they interact with space-time?

Cheers.