Say you try one of the other proposed approaches in the other answers and the screw is just stubborn and more of the shaft keeps snapping off...
Consider that you might not need to remove the screw at all. In a worst case, you can almost always snap off the screw close to the base of the wall (maybe a little grinding with a rotary tool like a Dremel to fully bury the head) and patch over it with some drywall joint compound like you would do for a screw hole anyway.