What cleaning product unclogs bathroom sinks?

a snake (it cleans out the drain)

No cleaning product will unclog a sink drain. It it's you bathroom, the clog is probably hair. A good product is Mr. Plumber in the grey bottle. If that doesn't work you may need a plumbers snake

I would use Drano. Follow the instructions on the bottle, then boil water and pour it in after the time it is supposed to be sitting. It usually opens mine right up. Use liquid Drano in a bottle. Those snake things will not go into my drains because they have an arrow-shaped thing on the end of them. I have one.

A plumber's snake. If you don't have one, a coat hanger might do the trick, depending on how deep the clog is and what it's made of.

Try your local hardware store, they have a plastic tool that you insert into the drain that will pull out the hair ball that is affecting the drainage. Seems to me that this device is called a Zip It, but do not quote me. Any way it is very in expensive and is something that you can do your self.

use Draino + rubber gloves