I am not a huge fan of having a white kitchen sink because it gets so easily dirty and stained. Trying to clean it with soap or cleaning solution doesn't really remove the stains.
A SUPER easy, natural and very quick trick is to use the trusted baking soda. Just sprinkle the sink with baking soda. With a damp rag/sponge scrub the sink and like magic the baking soda wipes away the stains. Like magic I tell you!
Wash the sink out and the sink is white and sparkly again.