Originally Posted by Andrew R Stewart

This can be a really frustrating event! You need to understand that at this point the shoes are at risk and there's no guarantee of saving them. First is details. What kind of shoe and cleat? If a shoe with accessible cleat screw plates then you have a lot of worst case solutions (grinding cleat/screw away and replacing plates). If the cleat is plastic then cutting/grinding away the cleat and leaving the screw to be grabbed with a vice grip is a possibility. If the cleat screw plates are actually "T" nuts that can spin within the sole then things get problematic. Unless you can somehow keep the sloe's insert from spinning the shoe is likely going to be tossed in the end. (I have heard of replacing the "T" nuts but not for years...). You can try various penetrating liquids like ATF and kerosene but they get messy and won't dissolve rust, just lubricate it.

Post more info and we can get more specific. Andy.