Porcelain and stainless steel sinks require different types of care. Porcelain responds best to chlorine bleach. Fill the sink with water, add half a cup of bleach, and let it soak for an hour or so. Before you reach in to remove the plug to drain the sink, be sure to put on a rubber glove. For severe stains, line the sink with paper towels, saturate them with bleach, and leave them there overnight. The exception to this rule is an old sink that's cracked or where the porcelain has worn through. The bleach can seep through to the metal backing and cause rust, which will make the stains worse. In this case, a gentle abrasive cleanser is probably your best bet. With stainless steel sinks, the biggest danger is scratching the surface. Baking soda is a safe cleaner that makes sinks sparkle. Rinse thoroughly and dry with paper towels to prevent water spots.
