I’d start with warm/hot water and a mild dish soap, and use a sponge with lots of elbow grease. If that doesn’t work, try mixing baking soda with a little water and gently rubbing the paste on the stains with a soft cloth or sponge. There are plastic-safe cleaners you could try, as well. The problem is that once the sinks get scratched, it can really make it hard to get stains out. Replacing with a stainless steel or other type of material is another option.
