Since you haven't gotten any replies, I thought I'd put in my 1 cent (not worth 2 cents).

I don't know if this would work for you, but I bought dh a screw extractor tool one year. It is really for screws that are broken off or too messed up on top for the screw driver to grip. I think it kind of works like a drill to get into the screw & then has enough torque to unscrew it. I think I bought it at Home Depot.

You could drill the screw head off, then use a plug cutter to remove enough wood from around the shank (with the dish already removed) to unscrew it with pliers. You could later use a longer screw to reinstall. Have you found a plating shop already? They're getting hard to find. The EPA has taken a toll on them.
Casey

Marti and Sombreuil -
Thank you for your posts. I asked a construction savvy man at work and he suggested PB Buster. It did remove the grout from the outside of the screw. I also bought a 3/32" screw driver with a handle. I did this because I couldn't get much leverage with the "eye glass" type screw driver. My wimpy hand and arm strength crushed the new screw driver head! Tonight, my dad also suggested the screw extractor tool. I'm not certain that it will work for such a small screw, but I'm certainly willing to investigate.

Yes I have found a plater. This 1957 very modern house, built high-end, had a really cool electric heater and a inset toilet paper dispenser. Here in Houston I found a plater and they did a beautiful job. We'll probably never use the heater, but it is a cool ornament. I have several recessed light trim rings and another toilet paper dispenser that is ready to go as soon as I can get these six darned soap and tooth brush holders off the wall. It will look really cool if I can get them re-chromed. If not, I'll have really cruddy looking ornaments around our sink. :) Again, we'll never "use" them as originally intended.

Thanks much for your suggestions. At least I still have hope.

Take care -
Jill

Try to find the right size screwdriver for the rest of the soapdishes and get one that has some length and a decent handle on it.