My sink is old but isn't scratched. The finish is dull and not even.
My quick fix is baking soda and vinegar. It fizzes, which is kind of fun, but needs to be spread all over and let set for awhile. I usually leave it and forget to get back before it dries, but I don't think that makes a difference. Just sprinkle it with water, and rub it around the surface, especially crevices and faucets. Rinse off. It takes off soap and hard water scum, which is a huge problem for me.
Just pour on the vinegar slowly. It'll make it own fizzy paste. Do not mix much at a time till you get the hang of it.
I like Bar Keeper's Friend.
Clean with Bar Keepers Friend. Coat with mineral spirits and let it sit for a while to soak in. Wipe off the excess.
I use what Delle said - baking soda then vinegar and water to get it clean. The next step is to use a vegetable oil on a soft cloth and rub it into the stainless steel in the direction of the grain. A lot of people don't realize that stainless steel needs to occasionally needs to be oiled - just like wood. After applying a thin coat of oil, use another clean cloth to polish the stainless steel. When you are done it will look like new - and you haven't used an chemicals. :) -Linda