This would be the actual wormhole. Incidentally, you'll notice that the embedding diagram looks a lot like a dip or a well in spacetime. I'll use the term 'gravity well' to describe the region of spacetime warped by the gravity generated by the presence of matter. This would be a great way to travel from place to place in the universe, except it is terribly unstable. The whole process of the wormhole forming and then pinching off sounds like it takes a while.
Most astronomical stuff takes a while.
Unfortunatly, the wormhole pinches off so quickly that not even light can get through. A static black hole simply isn't meant for convenient space travel. A charged black hole's singularity looks like this in the sense that the singularity is pointy and nasty, but it has the virtue of avoidability. Yes, the singularity of a rotating black hole is different than that of a static or charged black hole. The Kerr geometry describes it as a ring, whereas the other two are points.