[codethief]

Could you elaborate on your definition of volume here (are you talking about spatial volume or spacetime volume?) and how the curvature going to infinity at the singularity should imply its infiniteness?

My thought process here is the following: The inside of an (eternal) black hole carries four (Schwarzschild) coordinates t, r, theta, phi – r now being timelike and confined to the interval (0, 2M) and t now being spacelike and being any real number. That is, depending on when (at what time t) you cross the event horizon, you end up at a different point in space. The singularity at r=0 is then a point in your future which, like your own death, you cannot actually see but which you will nevertheless hit in finite proper time.

So in this sense I'd say the volume is very finite (if we disregard the (trivially unbounded) spacelike coordinate t which, as mentioned before, simply corresponds to the time of entering the BH).

[codethief]

I should add:

1. In the interior of the BH, the determinant of the metric is bounded since the Schwarzschild factors in the metric cancel out. So the volume measure doesn't do anything crazy as one gets closer to the singularity and boundedness of coordinates implies boundedness of the volume. Again, I'm disregarding the spacelike t coordinate because to me the relevant fact is that all matter reaches the singularity in finite proper time, so while we could theoretically stack lots of (actually, an infinite amount of) (massless) cubes inside a black hole, they would soon all get crushed.

2. Of course the situation is slightly different if we're talking about a growing black hole whose mass (and, therefore, radius) increases as we throw matter into it.

[lostmsu]

Unfortunately, I am not strong on mathematical part of GR, so feel free to disregard the rest of this idea, if math's the sole thing you are looking for.

A thought experiment: imagine we have a clock, falling into Schwarzschild black hole. Obviously, any real clock would have some non-zero size in all space dimensions. Here we will be concerned with just two: r and any orthogonal one. So for simplicity let the clock be a simple rubber-like oscillating ring with a fixed k and infinite resistance to tearing. (You could also take infinite k, but I'd argue that would not be physically meaningful in this setup)

As the ring is closing to the r = 0, its oscillations will slow down and come to a halt due to physical stretching along the r dimension. What I am trying to say is that maybe these oscillations make more sense as the measure of time for the ring people, than what a numerical value of proper time tells us. In a similar way the time singularity at the horizon is nothing special for a freely falling observer.

I am unsure how to interpret the fact, that the number of oscillations per proper time unit is going down though. Seems to be quite the opposite of my original note about the volume, yet something is ringing.

[lostmsu]

Unfortunately can't reply to the original question anymore. But here's another hypothesis I just read elsewhere (RU: https://don-beaver.livejournal.com/212422.html): the matter-energy falling towards "center" inevitably has some non-zero momentum, orthogonal to the direction along r. That means its fall is going to be (sort of) spiral, causing it to produce gravitational waves. The argument here is that the whole mass will be converted into gravitational waves before reaching the central point. Now what happens to those waves is a question still.