A lot of good suggestions here, I thought I might add a couple of additional ideas. First, broken screw extractors. Tools used to remove broken screws from watch plates after the screw head has been sheared off. Only useful with parts having a through hole, allowing the tool to grip booth ends of the screw.A detail view of the business end. These tools are scarce but the important feature is that the tool clamps down on both ends of the screw and then is turned to loosen and remove the broken portion of the screw. Modified C clamps or even a drill press could be used. Don't power up the drill press, hold the chuck stationary and turn the object with the broken screw by hand.Second, if your broken steel screw is in a nonferrous part (brass or nickle) then you can dissolve the screw in a concentrated alum solution. Heat some water and add alum until it you can't dissolve anymore. Then drop your part in. Heat and solution concentration all speed up the process. Make sure you have removed any steel parts you want to keep. This stuff will make short work of spring steel. My mentor demonstrated the technique on a clock plate but left the springs for the strike and chime levers attached. He said it wouldn't be in there long enough to hurt them and ended up making new springs. It won't damage but may discolor nickle and brass, experiment on any other metals before trying with an important part.Troy