Earlier quoted context omitted.
Or read Susskind's "The Theoretical Minimum: General Relativity". For a non-spinning blackhole at least, not only is the singularity not a point, it is a surface in time, not space (as the book explains, the space and time coordinates switch places as you cross the event horizon).
The way Brian Cox puts it, a singularity is a point in time: the end of time. I have trouble really conceptualizing black hole physics, I just think of it as a mass so great that nothing, including light, can escape its gravity. Works for me.
The singularity in a rotating black hole is entirely different but the interior of classical Kerr (rotating) black holes is one of the most controversial if inconsequential topics in theoretical physics because there are reasons to believe (without real proof mind you) the Kerr solution is unstable inside the inner event horizon so that whatever happens in there is not what that theory says.
And of course black holes are quantum objects which might actually have an “interior” entirely different from the classical picture.