Cut a fresh lemon in half and rub the cut side on the stain in your sink. Different stains respond to different treatments, so if the lemon doesn't work, there are other options you can try to remove the stains.
Leave the lemon juice on the stain and sprinkle on a little common table salt. The two elements together have a light bleaching action, yet it's non-toxic.
Wet the surface of the stain with water and sprinkle on a little baking soda. This works as a non-abrasive scouring powder. Rub with a damp soft cloth.
Place the stopper in the sink's drain and pour some white vinegar on the stain. Let it sit for 30 minutes. If the stain is even more stubborn, heat the vinegar on the stove top or in the microwave and pour the warm vinegar on the stain, again allowing it to sit for 30 minutes.