How do I remove a stain from a bathroom vanity sink?

If it's a standard porcelain sink you can use a pumice stone on it. It seems like it would scratch, but it doesn't. I first used one to get the stains out of the toilet in an old house we moved into. It even gets out those hard water stains.

I'm certainly going to try that. I've used all the cleansers, CLR , nail polish remover, vinegar, etc. I don't know what the stain is since it was stained when I bought the house. Thanks so much for your suggestion.

what type of stain so one can give a solution

A mixture of lemon juice and vinegar will remove hard water stains.

It depends on the type of stain and what the sink is made from.