VERY, very carefully. Years ago I stained my sink with hair dye. Well my roomie was having a party that eve so I hurried. I first used ajax then comet no change. Then I used amonia still stain remained. I used 2 other cleaners before it finally came out. Hours later during party a loud popping and cracking sound was heard from bathroom. The sink blew up! Shards of porcelain were in hall and everywhere! How embarrasing. That is my tale so be careful.
20 mule team borax, this stuff is amazing on porcelain. Sometimes you have to let it sit on the stain a while (overnight) but it makes your porcelain sparkle. Some stains will respond more quickly to baking soda and coarse salt paste scrubbed on the stain then rinse it off with white vinegar (it's also fun to watch it bubble) and follow the whole thing with a boiling water rinse. This will also clean your drain. If all else fails use a magic eraser from Mr Clean, that will get any stain off, but be careful using it on painted surfaces cause it will remove paint.
Once got it on the floor and did not notice till it was dried on. The Quick Blue hair bleach worked on the floor but I don't know if it will damage your sink.
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