Stainless steel sinks are probably the sturdiest of all sinks and if cleaned properly, can look great for a long time. Scratches are inevitable in a stainless steel sink with prolonged use, but the beauty of this type of sink is that you can usually scrub and polish the stainless steel to hide the scratches and improve the sink's overall appearance.
For minor scratches, mysinksandfaucets.com recommends to lightly scrub the sink with a soft scouring pad, like a Scotch Brite brand, and a cleansing agent, such as Comet or Bar Keeper's Friend (see Reference 1). With a dampened scrub pad and a little cleanser, scrub with the grain of the sink over the entire sink, not just one spot. Rinse thoroughly after scrubbing.
For deep scratches, use a pumice-rubbing compound found at a hardware or DIY store. Follow directions on the package and polish the sink until you have achieved the desired shine and finish (see Reference 1). Mysinksandfaucets.com warns about the importance of scrubbing and polishing with the grain and to treat the entire surface, not just a few areas, to prevent shiny spots on your sink (see Reference 1). If you polish the stainless steel too much, you may not be happy with the final appearance and you cannot reverse the process. Keep in mind that tiny scratches will continue to show as you use your sink but over time, they will blend in and not be as noticeable.
Another alternative for scratches is to use a nylon pad and an iron-free abrasive polishing compound, again, scrubbing with the grain over the entire sink (see Reference 1).
The previous scrubbing process will more than likely remove any rust and stains from the sink, but if not, or if you do not have scratches and only stains, they can easily be removed. Although stainless steel does not rust, hard water or other sources, like aluminium baking trays, will leave rust and stains in the sink. Mysinksandfaucets.com suggests using products like Bar Keeper's Friend, Gordon's Miracle Shine or Flitz to remove stains (see Reference 1). The sooner the stain is removed the better. Another choice is to use a solution of 10 per cent to 15 per cent nitric acid wiped on with a wet sponge, then rinsed (see Reference 1). Hard water stains can usually be removed with a 25 per cent vinegar and water solution (see Reference 1).
Companies like Carol Wright Gifts offer a "Cookware and Sink Scratch Remover" kit with refurbishing pads (see Resource 1). Chefscatalog.com also has a "Scratch Remover" kit, complete with four silicone carbide pads and polishing liquid (see Resource 2). The results of these commercial products should be fairly close to that of the process described by mysinksandfaucets.com for refurbishing a stainless steel sink and it appears the necessary items for the project are included. One of these all-in-one kits may be good to purchase for your project if you do not have a scouring pad and the correct scrubbing cleanser already at home.