Had half an 80 mm screw (5mm diameter) break off level with the surface of mortar between bricks yesterday. It was held in by a plastic wallplug.
Having wondered about using a screw remover, drilling it out, and several other suggestions on the internet, I thought about using heat. I have a soldering head for a gas-powered blow lamp, so heated that up and applied it to the end of the screw for a few minutes. After a while I pushed some narrow-nosed pliers into the soft plastic on either side of the screw and was able to grip it and turn the screw a little. After repeating this a few times the screw was out enough to grip it properly . After about 15 mm was out, I was able to pull it out with the remains of the plug.
Maybe this has already been suggested, and of course it would be of less use with a screw in a heat-sensitive surface. But it might prove useful if other methods fail.