Is it a vacuum? Matter? Nothingness? Do our models account for it or is it a complete mystery?
I suppose there would be some matter or energy (photons, anyway), anything that had just recently fallen through the event horizon, making its way from the event horizon toward the singularity itself.
An entire universe - maybe
There may be a “firewall” of superheated particles just inside the event horizon.
The simplest models have it all vacuum, everywhere inside and out (note that “everywhere” does not include the singularity, which isn’t a “where” any more). If you don’t make it a point to do something else, vacuum is what you’re going to get from the models by default. If you don’t mind a more complicated model, then you can put whatever the heck you like in there, so long as it’s spherically symmetric, because it’s all hidden behind an event horizon so you don’t care.
I think the answer depends on where you’re observing from.
If you’re falling in, the answer is, you and all the other stuff that’s falling in at the same time (and it takes forever to get to the center, even longer than a family car trip!) If you’re a ways outside, I believe the time dilation is such that anything falling in would get to the center in a jiffy, so even if lots of stuff is falling in all the time, it “looks” like a vacuum – one that you can’t see, of course, but you know the mass density is infinitesimal.
Chronos and other folks with actual clues about astrophysics, please let me know if I’m just full of it, which is highly likely. But I seem to remember reading something like this. No doubt even if I got part of it right, I’ve grossly oversimplified and misremembered. How’s that for fighting ignorance? And pardon the technical terms like “jiffy”. 
I think you got it backwards - if you’re observing from a distance someone else falling in, then they’ll seem to go slower and slower as they approach the event horizon.
However, if you’re the one falling in, the time dilation means that it looks fast to you.
One thing I’ve always wondered - if you’re closer to the black hole than some other, far-away observer, wouldn’t you perceive the even horizon as smaller than the distant person would? Seems like it to me.
CurtC has it right: To the infalling observer, reaching the center happens all too quickly, and if you do anything other than just passively falling straight in, you’ll just make it happen even quicker. But to the outside observer, you never quite reach the event horizon at all (though you will disappear entirely due to the redshift).
You don’t perceive the event horizon at all. There’s nothing special going on there locally; it’s only relevant if you consider the entirety of spacetime. There is (sometimes) something called an apparent horizon which a non-falling observer can perceive, but the infalling observer won’t.
Can’t you (indirectly) observe the event horizon by seeing where infalling stuff redshifts itself out of visibility? Let’s say there are a billion tennis balls falling into a black hole - wouldn’t you see the tennis balls that are closest to the event horizon as very redshifted, outlining where the event horizon is?
And if there are two observers, one very far away, and another hovering somewhere outside that event horizon, wouldn’t the inner observer see a smaller diameter event horizon, because he’s at a vantage point from deep within the gravity well?
Basically the event horizon is just the radius at which the escape velocity for the black hole equals the speed of light, so inside of that, there’s no observation.
I’d think the assumption would be that it’s just vacuum from there to the singularity itself, but we’ll never really know.
That’s what led to me thinking the way I do about it - somebody a great distance away can’t see within the event horizon because even light can’t escape out to him. But for somebody who’s observing from a position nearer the black hole, the light wouldn’t have to make it to the distant observer, just to the nearer one, thus some points will be visible to the nearer guy that are not visible to the far one.
I sorted this all out to my own satisfaction several years ago; however I realize that my gut feel for how it should work may not apply to these situations, so if I’m incorrect I’d like to know, and to understand why.
That can’t be right, because, as others have said, for a person falling in, the event horizon is imperceptible. There is no difference between a metre outside the EH and a metre inside. All the same stuff is present and correct - it’s just invisible to a stationary observer outside. If there was a sudden shift from “stuff” to “vacuum” then the EH wouldn’t be imperceptible.
I was assuming that it was the normal vacuum of space on the outside of the event horizon as well, so there wouldn’t be any change, since there would be vacuum on both sides.
If light travels from event A to event B along path P[sub]AB[/sub] and light travels from event B to event C along path P[sub]BC[/sub] then the path P[sub]AC[/sub] = P[sub]AB[/sub] ∪ P[sub]BC[/sub] is a path that light can take from A to C. Or in other words if the further observer can observe the nearer observe, he can also observe what the nearer observer is observing.
I concur with Shalmanese’s first question: “Is it a vacuum?” My theory shows this to be the most pluasible assumption. How energy gets conveyed from the event horizon to the “nucleus” of a black hole (or singularity) requires an understanding of the phenomenon of quantum mechanical “tunneling.”
Tunneling has nothing to do with it. The process by which energy gets from the event horizon to the singularity is known by the technical term “falling”.
smithddjrr: welcome to the Dope. Particularly this neck of it. This theory which is yours. What is it?
What lies between the Event Horizon of a Black Hole and its Singularity?
Rich creamy nougat?
Your daughter’s bookcase when she was a kid.
As others said, there’s nothing special about the event horizon except from the perspective of the distant universe. Locally, you can’t really detect it. From any location while you are falling in, there is some horizon from beyond which nothing can reach you, and from your current location, that place is special – but once you reach it, it isn’t special anymore, much like the ordinary horizon we see on Earth. You might think of that horizon in front of you as an event horizon specially defined for you. The official event horizon is the one specially defined for the surrounding universe in general.
The question “what lies there” has an implicit “now” built into it, and there’s a problem with that, because in general “now” isn’t well defined anyplace except for the location of the person saying it. That’s true everyplace and all the time, but it’s much more noticeable around black holes.
Since nobody can observe that region without actually approaching the black hole, a nice way to think of it might be that as you approach the hole, the specially defined event horizon for you is always ahead of you, and while things are exciting in your neighborhood, there’s still stuff around you, your baggage and whatnot. Before you get to the official event horizon, you see your locally defined version of it moving ahead already.
As far as what is beyond the effective specially defined event horizon that keeps receding from you, well, I think there’s a philosophical issue about the question itself, regarding whether you can really think of that place existing. But in practice you can keep chasing after it, and seeing your baggage floating tormented around you the whole way in.