What’s the best method of getting stains (coffee, etc.) out of a porcelain sink?
Any bleach-based cleanser should work.
Organic stains (coffee, wine, tea) respond well to hydrogen peroxide – I think they even make a super concentrated peroxide cleaners these days.
Coal tar derived dyes (stuff like hair color) can get into the little micro-cracks in porcelain sink and be real trouble, you need a special, extremely fine grit abrasive (NOT Comet or Old Dutch, too coarse) that you can find at a plumbing supplier (I think it’s called ‘Loxi’) and an orbital sander with one of those woolly pads.
I’ve never had any luck with rust stains
Leave a wet paste of bleach-based cleanser stand on the stains for several hours.
Or fill the sink up with water and put about an ounce of bleach in the water. Open several windows or you’ll stink up the house and then let soak for about four hours.
Rust stains and mineral deposits can be removed with Lime-Away or other acid based cleaner. DO NOT LET IT MIX WITH ANY BLEACH!
Wear eye protection (glasses are ok) and wash your hands immediately with plain water if you get either bleach or Lime-Away on them.
I have found only one product that gets out the most recalcitrant stains. It is called Zud. What Ajax paste and CLR could not remove, it did. Try all the other cheap recommendations first, then go buy some Zud.
Since you have porcelain, beware of scrubbing vigorously with any sort of grit cleanser. You will break the surface glaze of the sink’s finish and create a chromatographic surface which will capture and hold even a single drop of tea or coffee. Use Bon Ami or another grit-less cleanser for the job. If you use the Ajax paste, DO NOT scrub heavily afterwards. Avoid working with Scotch Brite unless it is well used already. The alumina grit in Scotch Brite is some of the hardest abrasive on earth.
Be sure to use hot water for this. The stain removing power of bleach depends on the 3rd or 4th power of the water temperature.
I use bleach but have to wash it off in a few minutes otherwise I get rust stains.