How to shine and clean stainless steel sinks?

I have a stainless steel sink in the kitchen which I am cleaning in preparation of moving out of the apartment we are in. It isnt terribly dirty and I first tried spraying lime away to clean it (particularly around the faucet area) but that didnt seem to do anything at all. Only softened the lime and mineral deposits which I then very carefully chipped off with a screwdriver %26gt;.%26lt;. Next I tried the mean green brand cleaning spray and it seemed to do a little bit better but now the sink has a dingy white "buildup" all over the rear areas. I am not sure what has caused it because while it looks like hard water marks. It is shiny and beautiful when wet but as it dries is starts showing white blotches all over and if you look closely you can see what looks like the sort of pattern of mineral buildup you might find on a shower fixture. I have heard barkeepers friend or something similar might work well? Does anyone have any input, I am at my wits end. No product seems to do anything at all! I have also tried vinegar to no avail. Thanks a ton for any help with getting this blasted thing clean! :)How to shine and clean stainless steel sinks?
go to flylady.net

she teaches everybody how to shine their sink
Many of you can't understand why I want you to empty your sink of your dirty dishes and clean and shine it, when there is so much more to do. ... Now if you have a stainless steel sink, I recommend all of the above directions with one extra instruction added: after you soak your sink, rinse well, and use SOS pads to scrub it.How to shine and clean stainless steel sinks?
Soft Scrub works really well.You may also want to try stainless steel.It's a spray,kinda like an oil.It will shine it up but it does leave a little bit of a oily residue.
KaBoom found at most stores.This cleans everything.Also try the Mr.Cleen Eraser