Alot of good ideas here. Here's my take on it. Have removed no telling how many broken fasteners out of just about every conceivable situation and the one I try to use if at all possible is drilling and using an extractor. Your hole must be centered and small enough to not let the extractor engage the the threads of the piece. The best bit is a cobalt 135 degree split point, run slow and plenty of cutting oil, it will drill virtually all metals. I use Snap-On cobalts. A Dremel tool is good for getting a small dimple in the center of the broken screw for starting the hole. The best extractors in my experience are the hex-head spiral type - a quality one, not HF or such. I have a Mac set. The typical square "easy outs" are nothing but trouble, I'd stay away from them. If all this fails, I would go with the drilling it out and installing a Heli-coil as mentioned already.