The tools for ******ed screw removal are the simply miraculous left handed drill bits.
No, this isn't an apprentice gag - they are exactly what they say they are drill bits that work in the opposite direction to normal. The miraculous bit is because they operate in reverse they unscrew the offending fastener as they drill in. No need to saw off the heads.
I have used stud extractors in the past and have also managed to snap one in the screw - then you are truly f*cked as you can't drill out the super hard steel.
The drill bits I've got are similar to these (I've not used this exact supplier):
www.ebay.co.uk/itm/251801698126
I've not had a ******ed fastener defeat me since I got them.