Since no one has actually peeked inside of a black hole we really can't tell for certain.
What we do know is that when we do the math on our models what we find are things approaching infinity. Sometimes these are just caused by using the wrong coordinate system, but other times when we change coordinate systems, the singularity still exists.
It's important to note that when speaking about infinity don't fall into the fallacy of treating it as a value. You cannot have an infinite amount of something, but you can have something which has infinite characteristics. Consider Hilbert's Hotel which is an example of the hilarity found when trying to add finite numbers and infinity together. The expression " + 1" is meaningless because you can't add a value to infinity any more than you can add "a + 1".
What's actually happening in Hilbert's Hotel is the addition of aleph numbers with finite numbers, which you can do, but has silly results. Aleph-0 + 1 = Aleph-0. But this just describes the extent of the set, suppose we took a sum and looked at it:
1 + 2 + 3 +
And no matter what you try to do with it, that extra one is still hiding in the sum. If you take this new set and subtract it by all of the natural numbers, you should be left with the result of 1. One of the most irritating things is when people say you can do things like you can in Hilbert's Hotel, writing it off like it's some quirk of infinity. But it's not. If you shifted all of the guests over to only even rooms, you would still have the same number of guests and rooms.
2((n) n) = 2 + 4 + 6 +
You've effectively just doubled the number of rooms. It's a sleight of hand that breaks the rules. "But!" you may say, "You have infinite many rooms, so of course you have a room at 2n!" If you do think this then you're still caught up thinking about infinity as a literal value. You don't have a room at 2n, your rooms only extend to n, and now half of your guests (which is still an infinite many) don't have rooms, but are left to stand out in an endless hallway.
In essence, one kind of infinity does not necessarily equal another kind.