I have been taught that space-time should be viewed as a fabric and that objects with a large gravitational influence indent that fabric. My question is, if the singularity of a black-hole punctures space-time, how is this accomplished if the universe is 3D? Can an object move completely around the black-hole in all directions? Would you be able to travel "below" a black-hole?
The 'fabric' visualisation only works if you ignore one of the space dimensions. For example, you can see that when an object orbits around another due to gravity (a motion in 4D), spacetime is like bending of a 2D rubber sheet by a heavy object placed in the middle.
If you throw a coin in, it goes around in the resulting dip instead of rolling straight, just like the in the real 4D case.
Space-Time is a 4-D structure which cannot be visualized. All analogies have limitations and your questions point to those limitations. Certain things can only be communicated effectively mathematically and space-time is really one of those things. Singularities related to black holes is a hot topic in research so there is no accepted right answer. But we can say that the known laws of physics come to an end at a singularity.