Not much. The elevator itself is actually not in orbit, so anything detaching from it immediately starts falling down, accumulates some funny velocity (it's falling down in vacuum, so it won't reach a terminal velocity as when falling in atmosphere) and fiercely burns up on reentry.
The Earth-side construction side might be damaged by the low-altitude debris but the high altitude part of the elevator (which makes up for about ~99% of it) should burn-up safely in the atmosphere due to high accumulated velocity.
On the other hand - blowing up a space elevator on a body with no or low atmosphere - you'll probably get a nice 1:1 globe with a finely engraved equator line.
I wasn't talking about the orbital pieces. I really don't know what would happen for certain to those. Large parts of it might just go flying off becoming a huge debris problem in orbit (like we don't have that problem already as well). I'm talking about the miles of construction in the atmosphere getting up there. That stuff isn't going to burn up and is going to land on someone. Just look at what happened when the twin towers came down. Now imagine 60 miles of that stuff coming down.
In computing, the mean time to failure keeps getting shorter.