Broken Wheel Locks:
Scene : Your Porsche Factory Wheel locks (the ones that take a 'key' and resemble a 'cap' are stuck (lost key..)
Solution : Center punch the hell out of the key hole until the 'guts' fall out, then using a mangler (screwdriver) mangle it until it comes off.
Broken Wheel 'Nut':
Scene: Some neanderthal that doesn't understand how dis-similar metals disagree used a high torque Impact wrench (your fault) on your Aluminum wheel nuts, OR: Some special needs adult installed the aluminum wheel nuts on the car without anti-seize, and after the years that have passed watching your tires dry-rot you decide its time to remove the wheel, breaking the nut so short that there is not enough remaining to use a proper 'tool' ..The HOrrors!
Solution : Get a 17mm Arbored Hole saw (you will have to remove the centering drill bit). Looks like this
Get 3 or 4 per wheel nut. Get some lubricating oil (light machine oil) and a heavy duty electric drill (put the recharging drill on the shelf this is real -drilling over here).
Carefully slide the hole saw into the hole. The goal is to erode the nut away without scarring the wheel, or damaging the stud. IF you are careful eventually you will erode the remainder of the nut away (use lots of lube, go carefully) to the point you can use a pick tool to get the metal remains out.
Use Eye protection. Metal shards blind people dumb enough not to wear protection everyday.
To prevent this particular form of self induced torture, use the correct antiseize on the ball cup end of the wheel nuts.
A little goes a long way.