Today we need an example of something weirdly-shaped and electrically conductive. There's no shortage of such things, so we might as well go with the iconic. This is the Statue of Liberty:
It's made out of copper, which over the years has taken on a decidedly not-copper color due to chemical reactions between the copper surface and the surrounding atmosphere. But it's still copper and thus a very good conductor of electricity. Unfortunately for our purposes here the statue is also hollow, and in fact the copper is only a few millimeters thick. This isn't unusual, almost all metal statues of any size are hollow. Metal is very expensive and very heavy. But for the moment, go ahead and pretend the Statue of Liberty is in fact a solid mass of copper metal.
Now apply an electric charge to it. Dump a bucket of electrons on it, fire up a Van de Graff generator, rub a balloon on your head and hold it close to the statue, whatever. Now there's an excess of charge on the statue. How does it distribute itself? We know that charges experience a force when they're placed in an electric field. Conversely an electric field is generated by the presence of charge. Thus the charges are going to be pushed around by their mutual repulsion until they reach a stable configuration. With a little thought we can figure out what the stable configuration is, even for something so complicated as the Statue.
It's not possible to quite do this in reality, but imagine that you want to very sensitively probe the electric field at a particular point. You do this by taking a single electron as a test charge, placing it at the location you want to test, and seeing which way it moves. In that way you can see what direction the electric field points.
But at this point we can also extrapolate backwards. Once we charge up the statue, there can't be an electric field anywhere inside. If there were, our test charge and all the other charges would be moving, which means they wouldn't have found their equilibrium positions yet. Once they find their equilibrium positions, they aren't moving anymore. Which means there's no field. But if there's no field, that means there's no net charge - because charge generates an electric field. So is there no charge in the statue, despite the fact that we just put it there? The answer is that there's no charge in the statue. Our argument shows that the equilibrium position of all the charge is on the surface of the statue. Any charged conductor will have all of its charge on its exterior surface.
This argument only works for conductors, since it requires that the electrons be free to move. In an insulator they can't, and so they'll mostly stay wherever they're put.
Back to the Statue. All of the charge is on the surface, but we have no guarantee that's it's evenly distributed on the surface. In fact it's not. In general the distribution on the surface will be a complicated function of the geometry. In particular it will tend to be highest at sharp points like the spikes on the crown, or (more saliently for other types of structures) lightning rods. And that's a story for another day - though I'm pretty sure that other day will be tomorrow!
I guessed correctly that it would accumulate in sharp points.
You may want to distinguish between in the statue meaning in the metal and in the statue meaning in the hollow core. How about going with a solid hunk of statue, like The Thinker, instead?
Charge resides on the surface of even hollow metal statues. I guess you can call the inner surface a surface, but that surface will have negligible charge.
And by the way, the point where the charge will be most highly concentrated are on those spike tips around the head.
I must have missed the lecture that explained why the charge accumulates at the sharp bits. I simply remember that it does. Why is that?
I don't doubt you, but your quick reasoning there is lacking at least one point in order to be convincing. You say:
Once they find their equilibrium positions, they aren't moving anymore. Which means there's no field. But if there's no field, that means there's no net charge[...]
Now, why is that argument not valid for the surface too? If the surface electrons are no longer moving, they're not creating a field and there's no longer a charge there either. Without some kind of extra explanation the same argument can be used to "prove" the non-existence of a charge at the surface too.
@5 - Now, why is that argument not valid for the surface too?
Because electrons at the surface are not able to move freely - they can't move outward.
I'm well aware of that, but that's only if the charge is place into the metal of the conductor or is outside of the conductor. If I stand inside of the statue holding a charge but not letting it come into contact with the metl, it will pull charge to the inner surface of the statue.