Tube Extractors are the Best Option
Sawtooth edges make these small tools very handy to have around the shop. Use a drill press, if possible, to bore a clean hole around the screw. Or, a hand-held drill with a guide block works fine for screws located outside of the reach of a drill press. Once the screw is out, patch the hole with a length of dowel or a matching plug.
For screw sizes #4 #8, use a 1/4″ tube
For screw sizes #4 #10, use a 5/16″ tube
For screw sizes #10 #12, use a 3/8″ tube
Tips to Avoid Breaking Screws
Use a better quality screw.
Drill properly sized pilot holes, which are different for hardwood and softwood (see chart at below).
Lubricate the screw with wax or paraffin. Grease or oil may leach into the wood causing finish problems.
Always clamp the workpiece to your bench or in a vise to hold the piece securely.
For brass screws, first install a steel screw of equal size to thread the hole, then replace that screw with the brass screw.