Originally Answered: What's up with the valve for my sink?
The screw on the top of the handle normally just holds the handle to the valve stem. If you remove the handle and find that using a pliers or vice grip pliers, you can turn the water off, then your problem is limited to deciding how to solve the problem with the handle. But as you noted, if the screw is moving with the handle, its more likely the problem is in the valve stem or the seat.
If the handle is loose and you can shut the water on and off with the pliers, then look at the top of the valve stem and see if the ridges that run around the top of the stem have been stripped, or if the problem is inside the handle. If the ridges are stripped smooth on the stem, get replacement handles that provide a little metal cube along with the handles that attaches to the valve stem with an allen wrench (usually included in the package) that you can tighten the screws in the cube to the stripped end of the valve stem. You might want to file a portion of the stem flat to hold the cube on tight. Then you can screw the replacement handle into the cube to hold it on the stem.
Assuming the end of the stem was okay, if the inside of the handle is stripped, just get a new handle that matches the type of stem you have and you should be good to go. If the handle isn't stripped either, the problem is most likely either the washer or the seat that the washer closes against.
If you have water shut off valves under the sink, shut off the hot water line. If not, find whatever valve you can working back to the water line that comes in the house so you get the water shut off to that valve. Loosen the nut that holds the valve stem to the faucet assembly until it slips off and allows you to pull out the valve stem. It might be an old style stem with the bottom having a rubber washer held on by a screw. If so, replace the washer with the same size and shape washer, making sure the new one fits inside the ridge that runs around the outer edge of the bottom of the stem. Look inside the hold the stem came out of and note the surface the washer hits when the valve is turned off. is it smooth or pitted/irregular? If its not smooth, you'll want to try to remove the seat. Most will come out with an allen wrench or a flat blade screwdriver of the correct size, but shine a light in there and get a look and use your judgement as to what is most likely needed to twist it out. It has threads and will come out with enough torque applied. If you can't get it out, you can get a hone at the hardware store that will allow you to smooth it out enough so the washer will seal against it and hold the water back under pressure. If this all sounds like a lot of trouble, and you would prefer to buy a new faucet, you can certainly do that, just shut off the water to both valves and look under the sink for what you need to detach to remove the faucet.
If the valve stem included a cartridge with o rings instead of the traditional washer, you can either purchase a new cartridge at a good hardware or plumbing supply store, or you can just replace the o-rings on the cartridge. Take it in with you to find the right parts.
Then just reassemble in reverse order and test with the water back on.