Both Black Holes and Wormholes are conjectured concepts, in that because we can't see beyond the Event Horizon of a black hole, we must follow the equations of General Relativity, rather than actual experimental data or direct observations.
In the case of black holes, experimental data such as the recent LIGO Experiments, gravitational wave detection seems to greatly strengthen the case for the existence of black holes, as they conform to theoretical predictions regarding the strength and properties of gravitional black holes, made years before we could test them. In addition, General Relativity has not failed any experimental test of it's predictions, across a wide range of modern physics, from GPS receivers to deviations of the orbit of Mercury.
Having said all that, we cannot hope to obtain experimental date on what the Singularity of a Black Hole actually is. Because light cannot escape, we are limited to making (hopefully) educated guesses as to what properties if any, a region of spacetime compressed to infinite density (a singularity) possesses. At the center of a black hole lies the singularity, where matter is crushed to infinite density, the pull of gravity is infinitely strong, and spacetime has infinite curvature. Here it's no longer meaningful to speak of space and time, much less spacetime. Jumbled up at the singularity, space and time cease to exist as we know them.
One proposed explanation is that the singularity may act as part of a wormhole.
Lorentzian wormholes known as Schwarzschild wormholes or Einstein–Rosen bridges are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, and that are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the space-time should not have any "edges": it should be possible to continue this path arbitrarily far into the particle's future or past for any possible trajectory of a free-falling particle (following a Geodesic in the spacetime), unless the trajectory hits agravitational singularity like the one at the center of the black hole's interior.
Embedding diagram" of a Schwarzschild wormhole
We are trying to represent 4D spacetime on a 2 D page, so please don't take these diagrams as more than a general outline.
In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black hole region can contain a mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see light that fell in from the other one), and likewise particles from the interior white hole region can escape into either universe.
I would recommend that you read the Wikipedia articles that I have linked above, for more details on these conjectured distortions in spacetime, caused by extreme gravitional effects.