Step 1: Soft Abrasion
So back to it, I don't like to pass strong chemicals to the waste water system but I also don't like stains on my sink.
I have found that traditional nylon scouring pads are not tough enough, but I always have aluminium foil in the kitchen.
Take a sheet, scrunch it into a loose ball and clean in a circular motion with the water running slowly. You may need to make subsequent scourers as the ball will tighten and become a smooth rounded ball, when this happens it is no longer effective.
Step 2: Result
I have also used this method for cleaning machine parts, just always ensure what you are cleaning is a harder metal than aluminium and your good.