Spray glass cleaner over the entire stainless steel surface, paying special attention to any spots. Spot-cleaning stainless steel will make the areas you cleaned more visible; cleaning the entire object prevents this.
Wipe the glass cleaner off with a clean cloth. Buff the spots off the stainless steel with the cloth.
Apply a drop of liquid dish soap to the stainless steel surface if the spots are still visible. Work the soap into the stains with a wet cloth. Apply pressure to the areas around the spots until the spots are gone.
Apply white vinegar to a cloth and rub it into the stains if you can still see them. Work until the stains are gone. Wipe the entire surface down with the cloth to ensure that the spots you rubbed don’t stand out.
Rinse the stainless steel with clear water, and polish it with a clean cloth, rubbing in the direction of the metal grain.
Apply a dime-sized amount of vegetable oil to a clean cloth. Rub the cloth into the stainless steel in the direction of the grain. This will help polish the stainless steel. Buff off any excess oil with another clean cloth.