The screw heads have rusted off.

You will have to drill them out.
Start with the smallest drill bit you have. A 1/16 would be best. Try to get as close to the center as you can. Drill about 1/4" deep. This will be your pilot hole. Then increase the drill bit size slightly and repeat.

Then increase the drill bit size to the size of the screw head. Repeat again, drilling only deep enough to cut away the screw head.

Remove the faucet handle. Hopefully you will have enough of the screw sticking out to grip it with a small pair of vice grip pliers

If not, you can try drilling out the remainder of the screw and re-thread. Chances of success are slim but you can then replace faucet stems.

I would start as Harold suggested. But one more detail: Drilling is fine, just make sure you don't damage threads on the inside of the stem. If you manage to damage the threads, you will have to replace the stem.

If it is really that badly corroded, I would break the handle around the screw. Then it is easy to grab the end of the screw as you will have at least 1/4" of screw sticking out. Of course, you will have to replace the handle

Get new Stainless Steel or Brass screws. Looks like the corroded ones were not one of those. Moreover, dip the screw into Plumber's Grease before you screw it in

Hope that helps

Milo

Hello MRBEES5578

You have two threads open I responded to your 1st thread yesterday, but both experts gave you excellent advises, good luck.

John

If you don't have a finger (die) grinder
http://d2blls2j5typlg.cloudfront.net/media/catalog/product/cache/1/small_image/245x/9df78eab33525d08d6e5fb8d27136e95/i/m/image_8421.jpg
A drill bit the size of the screw head can be used to finish eliminating what ever is left of it and the knob should pry right off. If the screws are rusted, then expect to use some elbow grease, and leverage to pry the knob off. I would also have some type of joint breaker, like WD40, handy. A puller works best

http://c.shld.net/rpx/i/s/pi/mp/7840/prod_8013957820?src=http%3A%2F%2Fproductimages.spa ce%2Fproducts%2FCapri%2FCP10505.jpg&d=d9bc1013abe17f508765c044ef70d23a6507e491&hei=245&wid=245&op_sharpen=1&qlt=85

Once the knob is off, removing the screw remains is easier.