It is almost impossible for a black hole to collapse without containing some angular momentum so the only valid solution is the Kerr black hole.  This describes a collapse inside a more complex event horizon towards a ring "singularity" containing the angular momentum.  A rotating black hole of a given mass also has a maximum limit of angular momentum any excess of angular momentum over this limit must be expelled before any further collapse can take place.  This contributes to the bipolar jets associated with quasars and supernovae.  Now the detailed processes involved in collapse inside the event is largely dismissed because the end result has been established.  I feel this is a bit like saying that our own universe is uninteresting because thermo dynamics and physics clearly shows that the end result of our universe is the "heat death" which is a lot of isolated black holes of various sizes very slowly evaporating in vast  isolation over a period of time hundreds of orders of magnitude longer than the universe has been around already.  The truth is that what is interesting is processes that go on (including life and us) on the way to this cold isolated death.  Let us therefore consider what might happen inside the black hole even though we will never be able to go there and observe it.

Firstly. let us consider the collapse inside the event horizon of a simple non rotating black hole.  To keep things simple let us assume that no further material is falling in for the moment.  The Penrose diagram shows clearly that all roads lead to the mathematical singularity at the centre i.e.  All the light cones end up on the singularity I agree about this but let us consider the process in more detail.  This process will take some time  (at least the time taken to travel at the velocity of light from the first event orison to the central singularity)  this may only be a few microseconds but a lot of things can happen in a few microseconds as our models of the early stages of the big bang clearly show.

Now the first formation of the event horizon is when light cannot escape from the hole to "infinity" but close to this horizon it may not escape but it can travel a certain distance before it is dragged back by the gravitational field.  particles can only travel a shorter distance depending on their energy but they can still travel. Particles and radiation will still be interacting and some of them will as a result of these interactions be ejected as far as possible from the hole before falling back  this process adds to the time taken to collapse to the singularity.

This is work in progerss