Have you tried Bar Keepers Friend? Sold in the cleanser aisle.
You can use a powdered cleanser or a brillo pad. The rust will return if it has eaten into the sink . To stop it, get some epoxy and cover the rust spot.
Always clean stainless with baking soda. Great for kitchen sinks and cheap. I leave under the sink as order absorber too. Did you know you can use sand paper on stainless? If you need a fix that will not scrub out with cleansers then use sandpaper and sand with the grain of the stainless. I wouldn’t do it on a newer sink because they put clear coat sealers. FYI got the info from my plumber. I damaged mine with chemical.
Bar Keeper's Friend is the best cleaner for stainless steel. We had a family emergency and had to be out of state for a week. I left a metal cake pan (old) in the stainless steel sink and it rusted terrible. Bar Keeper's Friend got it right out. You can find it where comet or ajax cleansers are sold.
Thanks to all of you for your answers. As it turns out, I had Bar Keepers Friend already. I was trying to use up my Comet before I opened it!
Isn't it wonderful to have a site like this!?