My bathroom sink is a real pain and clogs all the time. >.<
I prefer to use things that are safer for my family, so I don't buy harsh chemical products when I can avoid it.
Use a plunger to drain the water out of your sink as much as possible,
this may take some time, so remember Patience is a Virtue. ;-) Once
your sink is as empty as you can get it, pour some baking soda down the
drain, maybe about 1/4 of cup or less. Be sure you have a fan or other
ventilation working. Then pour some vinegar down the drain on top of the
baking soda, probably about the same amount as the baking soda or a
If your drain still isn't clearing properly, repeat the process over again until it is.