a tightly screwed nut…don’t look at me like that, I am only passing instructions to the ladies.
Force. There are several ways to increase the amount of force, or torque you can apply. First of all, be sure you have the correct size screwdriver. A screwdriver that is too big or too small can strip at the head of your hole. If the screw head is sticking up, you may be able to grasp it with vice grips, and turn it that way.
If the screw is not sticking up, try placing the screwdriver in the slot and pressing down while grasping the upper, square part of the shaft of the screwdriver with your vice grips. You may be able to apply enough force to turn the screw that way.
Finally, even though it sounds counter-productive, try to tighten IN, the screw. If you can move the screw at all, move it back and forth, again and again and eventually it will come free.
Impact. Be careful that the work area surrounding the screw will take the impact gently before trying these methods. Occasionally, a wrapped hammer inserted in the slot is enough to break in as long as the adhesion is ample to allow the screw to turn.
A better method yet is to hammer in the screwdriver while you are turning it. You can buy a gadget called an impact screwdriver for this purpose, as well. It is designed to rotate the screwdriver bit when the handle is hammered in. This works because the tip is being forced into the slot while turning, back and forth, again and again.
Last resorts. If none of these methods work, and you simply HAVE to get the screw out, there are a few other things you can try. These things might temporarily destroy your work area, but should leave the screw hole intact so you can replace it with a new screw when the need arises.