I don't write myself into a corner very often, but when I do, it's usually fixed by finding a loophole in a very obscure area or exploiting a Shrödinger's Gun scenario.

When I get myself into that kind of fix, the first thing I do is re-read what I've written up to that point (or at least the last few chapters). Often this is enough to get my brain working again.
The second thing to do is think about where you want to end up. What's keeping you from getting there? How can you fix it?
If none of this works, I'll often switch to writing something different but similar for awhile and just let the problem roll around in the back of my mind for a week or so. Usually after a few days the creative juces start flowing again.

depends on what makes you feel more comfortable. I usually leave the story for a while (a month, more or less) then re-read it and try to find the problem in order to fix it. In my experience, corners appear when you haven't led your characters the way they wanted to go, so you just have to walk back and take a new path. More or less what everyone else said.

The thing about corners is that usually there's a reason or another why you can't get out, and you were forced into the corner because of loose ends being tied up.

I usually cut off the corner and start again where I began to get railroaded into the corner and try and take it in another direction.

Kind of like knitting. Make a mistake? foot of your sock too short? Rip the sock out and go back to the problem. Redo it the right way.

This is what drafts are for! :D

Or rather, sit muttering solutions to myself and drawing mind maps until I find a way out of the problem.

I usually start with a good idea, and end up at 1/3 (or 2/3 if I am lucky) stuck usually with "and now what I do?". However, I leave the story alone for a bit, and resume hen I get a good idea. It's time-consuming (a Knight Rider-Card Captor Sakura crossover was very close to be considered dead fic since it once took six months to write a chapter, but I also had a dead computer in the meanwhile. If you are wondering, i's in Italian, so I don't think linking it would be useful at all), but it works.

For me, writer's block is a sign that I did something wrong somewhere. It can be anything, really--someone said something they wouldn't have, or didn't do something they should have, or maybe I just didn't understand their motivations properly, or whatever. I go over the piece until I figure out what it is, fix it, re-write whatever needs it, and go from there.

~Neshomeh

I put the story aside and pray I find a way out. In the meantime, I move on to other stuff. It's worked once. Meanwhile, I'm sitting on a 20+ unfinished page story that is still unfinished.