I am trying to unscrew the screw head you see in the picture. It is from the kitchen faucet handle.
From looking online, it is usually a hex screw head. I tried a hex-key wrench, it didn't work, and tried every tool I have that could fit with no luck.
I am clueless, which one to use. Appreciate any help. Thanks!
It appears that the hex sides of that screw head have been rounded off due to the use of an ill-fitting hex wrench. A metric and an SAE hex wrench will both fit in there, but the use of the wrong one, while it feels like it fits properly, can be just loose enough that if the set screw is corroded in a bit, the wrench can slip and round over the edges.
You will need a screw extractor, like this:
Image courtesy of Lowes.com. No endorsement of brand or vendor intended or implied.
You insert this into a drill and run the drill in reverse. The extractor has a reverse thread that drills into the head of the screw and also turns the screw to back it out.
For the small added expense (and probably near impossibility of finding them individually), I'd suggest that you purchase a set of 3-5 extractors. That will help ensure you have the proper size for this screw, and you'll have a variety for future use. Note that these should work on nearly any screw with any type of screw head - slotted, Phillips, Torx, or other - not just the hex head of your set screw.
Once you have it out, you'll need to take it back to the hardware store/home-improvement center to buy an exact match (thread & length) replacement. As much work as you put into getting it out, you don't want to reinsert it and have to do it again...
I mostly agree with FreeMan, but I think this was originally a Torx (six-pointed star) drive. The corners look deeper than I'd expect from an Allen drive.
I would get a set of Torx bits and look for the largest one that fits (which is the correct size for any screw head). Gently tap it in with a hammer, then attach the screwdriver or ratchet handle. While keeping the bit in line with the screw and pressed in tightly, see if you can turn it.
Once it's out, replace it.
Find the largest allen or torx wrench or bit that is small enough to fit in the rounded-off hole. Mix up some 5-minute epoxy or epoxy putty, and put a blob of it into the hole. Immediately insert the allen or torx into the epoxied hole and let it harden for 24 hours. You now have a perfect allen or torx cavity with a tool in it. Unscrew carefully. Replace screw with a new one.
Giving a cheaper option, maybe you can try to cut a line with a rotary tool and simply use a flat screwdriver
