Bleach is great for my sink too! It's crazy how fast the sink can turn ugly on you, when you're not paying attention. Mine developed some seriously gnarly stains.
I filled my sink with water and about 3/4 cups of bleach and let it sit for about 5 minutes. Equipt with a heavy duty glove, I drained the sink and it's stains.
I noticed there were a few darker lines still lingering on, so I scrubbed those with some comet and voila'!