I shake Arm & Hammer baking powder in it and get a wet paper towel (haha.... mine has to be a Viva paper towel) and I polish it up. Then I completely rinse the sink.... sides included and dry it with one dry paper towel. (There are SO MANY uses for baking soda. Go to armhammer.com for more of them.)
My sink appears to be clean before I actually start to clean it, but I was surprised the first time I wiped and looked at my paper towel. It was a brownish color. (I don't know if that's from pouring coffee down the sink sometimes or just because the sink was dirty.)
BTW- I use a spray (can) that I bought from Lowe's called Home Remedy Plus Stainless Steel Cleaner Protector for about $6 to clean all my stainless steel appliances. Sometimes I'll spray a bit on a rag specifically for stainless steel and rub it around on my sink, too. I'm very happy with this product.