Theoretically these should always grab when used properly. Generally they come with a kit that has the correct size reverse bit and the associated drill bit. Edit: Just checked and harbor freight sells a whole set of these for like 10 bucks.

You'll want to use the regular, smaller bit to drill a hole into the screw, then (generally with a tap wrench) use the reverse bit to grip on the inside of the hole you drilled. Just back out the screw while the bit is trying to go deeper. I've never had one of these fail on me.

Hardest part is getting the initial drilling straight on the screw, but there is quite a lot of tolerance.

Also, lube up the bit while youre drilling, WD40 works fine for this. And blast the screw with PB Blaster before unscrewing it.