A "singularity" is a mathematical thing that belongs to the domain of a function. It can be a point, a line, a circle, a figure of some sort, a plane, a volume, etc.
Most often, unless you have a degenerate function, a singularity's dimension is at least one less than the space in which it exists.
A "black hole" is an object with an escape speed greater than the speed of light. In normal physics, we cannot know what the nature of that object is, only its basic characterisitics (mass, maximum radius, rotation, charge).
In astronomy, it is normally represented by its event horizon (with a finite radius as seen from the outside) because nothing from inside the event horizon can be perceived. Light photons can not escape from inside, even if they are moving at the speed of light, because from any point inside the event horizon, the escape speed is greater than the speed of light.
For "small" black hole, the required density is so high that we can't even imagine what the "thing" might be. We know only that it cannot be ordinary matter.
When cosmologists (people who apply mathematics to the behaviour of space itself, as if space were an object) try to solve the functions that describe the behaviour of space, they hit a singularity whenever conditions exist that would represent a black hole, in the real world.
A singularity is the place, in the "input" of the equations, where the output grows without bound (what ordinary people call "infinity")
For a "small", non-rotating, non-charge black hole, this mathematical singularity has the shape of a point.
However, we do not see how a non-rotating object that massive and that small could be non-rotating in our universe (especially since there is no real "rest" reference against which to measure non-rotation).
For a rotating, non-charge black hole, the singularity has the form of a circle.
And so on for other types of black holes.
This mathematical singularity may have nothing to do with the actual shape of "whatever is inside", but we do not know, because we do not know how things work inside the event horizon.
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Newton showed that for orbital calculations (objects in orbit around Earth), we can mathematically consider Earth's mass as being in one single point (Earth's centre). And it works. However, this does NOT mean that Earth is a single point. It is a sphere with a volume, and mass spread out everywhere in that volume.
The singularity in a black hole is the same thing: a mathematical trick to make calculations easier.