What is the size of the singularity?
1. Is it 0 cm?
2. Is it below Planck length?
3. Is "the size of singularity" the wrong question, such as asking "what is the length of 50^{0} celcius"?

What does that mean?
From event horizon all the way, just before, the centre, is vacuum?
And suddenly there comes the singularity?
In mathematic:
[itex]\frac{2}{0}[/itex] is twice as much as [itex]\frac{1}{0}[/itex]
What about this singularity?
Is the singularity of a black hole 20 solar mass is twice as "big" as the singularity of a black hole made of 10 solar mass?

The singularity, strictly speaking, is not part of spacetime, and it doesn't have a "size".

It means that, as you pass through 2-spheres of smaller and smaller radius inside the black hole (i.e., as the ##r## coordinate goes to zero), the spacetime curvature increases without bound. Saying that spacetime curvature is "infinite at the singularity" is really a sloppy way of saying that spacetime curvature increases without bound as ##r \rightarrow 0##.

Yes.

No. As I said above, the singularity is not part of spacetime. In the classical GR model of a black hole, an object that falls inside the black hole is destroyed by the spacetime curvature increasing without bound. So nothing ever actually reaches ##r = 0##.

No, both expressions are undefined, so it's meaningless to ask what their relative sizes are.

The singularity is essentially a mathematical fiction. It can't exist because it's somewhat like dividing by zero.

Within General Relativity, you have to remove the singularity from the part of the space-time you're describing, or else your equations become nonsensical. To really understand the interior of a black hole, we need to know the correct theory of quantum gravity (and even then it may be difficult to be sure). We don't know right now whether quantum gravity is important to understand the interior of the black hole just beyond the horizon, or whether it's only important when you get close to the center.

So?
After supernova explosion, the star just disappears from our universe?
Leaving only its mass in our space time?
Is it like 4D object cut through our 3D space?

Thanks PeterDonis, I remember you answered me for Twin Paradox symmetry/asymmetry. It seems I get invaluable help from you.

Yes and no. As nothing inside the horizon of a black hole can escape, anything within the horizon is, in effect, outside of our universe.

The point PeterDonis was making is slightly different: in order to make the equations make sense, it is necessary to carve out a little hole around the center of the black hole, label that hole, "Here be dragons," and do not attempt to describe what goes on there. This is General Relativity telling us that it is fundamentally incapable of describing the very center of a black hole.

As I remember weeks ago. I'm doing a Schwarzshild calculation. For a black hole 1 giga solar mass, the Schwarzshild radius is around 10 000 AU.
Are you trying to say, that an object inside this big radius minus 1 cm is outside our universe? What universe do you mean, the observable universe?
Do you mean that "Black Hole" is literally a "hole", I don't know about "black".
A sphere with 10 K AU radius is outside our universe?
And an astronout falling inside Schwarzshild radius still can see our "universe", though somewhat terribly curved but we can't see the astronout, only the astronout state just before it goes to event horizon.
So?
An astronout "outside" our universe can see our universe?

Anything past that horizon cannot be observed. Whether you call that in or out of our universe is a matter of semantics I don't care to worry about.

Not for long. If the interior of the black hole were accurately described by General Relativity, the astronaut would pass into the region containing the singularity (the region that we can't describe with General Relativity) within a very short amount of time. However, we don't quite know how quantum mechanics changes this picture: it may only become relevant at the very center of the black hole, or it may become important at the horizon itself. For example, in General Relativity, there nothing particularly special occurs when the astronaut crosses the horizon. But there has been some suggestion recently that the infalling astronaut would be fried by a "firewall" upon crossing the horizon: http://en.wikipedia.org/wiki/Firewall_(physics)

Edit: I looked up the maximal lifetime of an object falling into a black hole, and it's approximately ##10^{-5}s## times the mass of the black hole in solar masses. So for your hypothetical ##10^9## solar mass black hole, it would take just under three hours to reach the center from the point of view of the infalling observer. This is assuming General Relativity is accurate for a good portion of the interior, of course.

According to classical GR, once the collapsing star forms a singularity, then yes, the matter and energy in the star disappears; all that is left behind is its gravity--the spacetime curvature that was originally produced by that matter and energy.

As Chalnoth pointed out, though, what this is really telling us is that classical GR is no longer valid once the spacetime curvature gets too large. So at some point inside the black hole, the classical GR description of what is going on must break down.

No; I'm saying that, at some value of ##r## which is greater than zero but less than ##2M## (the radius of the hole's horizon), the classical GR description breaks down. For a hole with a mass around the mass of the Sun, we would expect this value of ##r## to be much closer to zero than to ##2M##, so the classical GR description would work well inside the horizon. But we won't really know for sure until we have a good theory of quantum gravity.