Wash the entire sink interior and rim with hot, soapy water. Use a nonabrasive cleaning pad to dislodge heavy dirt residue.
Soak a cloth rag in pure white vinegar. Lay the rag over the bottom of the sink. The vinegar breaks down hard water residue that can cause dullness. If white residue from hard water exists around the faucet assembly, soak a second rag in vinegar and place it over this residue. Alternatively, use a commercial lime removal product.
Soak the sink with the vinegar for 15 to 20 minutes. Remove the rags and scrub the softened hard water spots with the cleaning pad. Soak and scrub stubborn spots a second time if they are still visible.
Dampen one corner of a clean rag with lighter fluid. Scrub any rust spots with the lighter fluid to remove them, then rinse with cold water. Rust sometimes occurs in deep scratches and dents.
Wipe away any dark discolorations with a rag soaked in club soda. Dark rings occur when you set a hot pan in the sink and the heat discolors the steel.
Apply a commercial scratch remover formulated for stainless steel to badly scratched sinks, following package instructions. Removers work by buffing away the scratch with a mild abrasive and leaving behind a polish that helps to fill and disguise deeper scratches.