I've always had excellent results using a product called "Whink." It's sold through Ace Hardware stores throughout California. My Mom always had a bottle on hand, and it's amazing for removing rust from pretty much anything - ANYTHING! Fabrics, too! On the chrome drain, you might need to use a super fine sandpaper or steel wool, too. Good luck!
You can replace the whole thing for not much money. It just unscrews. Amazon has several for under $10; so does Walmart or just about anywhere else. Look at youtube videos for "replace bathroom sink drain."
Use borax and lemon juice Make a paste by mixing borax and lemon juice and apply it to the rust stain. Allow the paste to dry and then rinse it down the drain. Try cream of tartar Make a paste by combining a mild scouring powder, cream of tartar, and hydrogen peroxide. Apply the paste to the rust stain and let it sit for 30 minutes. Then rinse the paste away. Use TSP Dissolve 2 to 3 tablespoons of trisodium phosphate, or TSP, in a gallon of water, stop the drain, and pour the solution over the stain. Then scrub the rust mark with a sponge or rag. When the stain is removed, thoroughly rinse the sink and fixtures. se oxalic acid Combine one part oxalic acid to 10 parts water. Apply the solution to the rust stain and then thoroughly rinse. Use a pumice stone Run warm water over a pumice stone so that it absorbs some of the water and softens. Then scrub the rust stain with the stone. Use gentle pressure, but be careful not to scratch your sink. DO NOT USE BLEACH, IT WILL ACTUALLY SET THE RUST STAIN. If the stain comes back, you will have to replace the drain assembly.
Spray with vinegar let set a few minutes and wipe clean.
Agree with An re Whink rust remover. It's often sold in the laundry aisle. Walmart carries it. It's incredible on white porcelain surfaces - sinks, toilets, etc. However your sink drain looks as if the surface might be damaged. In my experience it might be easy to replace or it might be an ordeal. I replaced the drains in my kitchen sinks a few months ago in order to match the oil rubbed bronze faucet I'd just installed. The drains were rusted in place and it took quite a bit of work to eventually loosen them enough to get them out. When I finally did wrestle them free I found some rust was also present on the white porcelain of the sink drain opening. That was not surface rust but damage where the porcelain coating on the ancient Kohler sink had worn off and the iron underneath was rusting. I sanded all the damage off and then repainted the drain opening with white oil based touch up paint. The area would be hidden under the new drain of course, it was a structural repair, not a cosmetic one. I needed to prevent the rusted area from spreading. In any case, you might try removing the drain to see if it comes out easily. If so, I agree with the other Julie M. above, replacing is probably a better long term fix. I did run into a similar problem with one of our bathroom tub drains. I was not able to replace it and it was visibly tarnished, but structurally intact. I sanded it by hand and then placed a cardboard box with a hole (big enough for the drain area) that I'd cut in the bottom to protect the tub. I used painter's tape to seal the area around the hole completely and then spray painted the drain in place with one of Rustoleum's metallic spray paints. I gave the drain several coats after waiting for it to cure in between for the recommended amount of time - usually 24-48 hours. The paint finish on that tub drain has held up well but this is just an interim fix of course, not a permanent solution.
It's rusty because the chrome finish has failed. Most anything you do at this point will only be a temporary fix and you can expect the rusty areas to spread because of the constant presence of moisture. Chrome is applied to metal using electroplating. Not a job for anyone except a professional. However, that said, I camouflaged rust on my footed bathtub drain using a tube of silver Rub N Buff. It is waxy and will stick to that metal pretty well. It won't take much and one small tube will probably last for years. Buy at any hardware store or craft store for just a few dollars.
T cut used for car body work, removes rust and brings chrome up shiny and new :D
Try the powdered Barkeepers Friend. Dampen the rusty area, cover it with the BK , allow it to sit a few minutes, then scrub the area with a non-abrasive sponge or similar tool. I have removed rust from a variety of surfaces using this product, which is available in most grocery and variety stores--very inexpensive, too. I have had much less success using creamy version of Barkeepers Friend. Suggest you stick with the powder.
Once rust takes hold on chrome with constant water exposure, you can temporarily keep it at bay, but eventually it will come back. You can buy a new sink set at any big box store and simply replace the rusty part. If the rust was on the actual sink basin, then it can be fairly permanently fixed using epoxy putty but on the chrome....best to just replace it. No plumber needed, just buy the kit, pop out the old and pop in the new. Done deal. FYI: It's best to pop out the old one & take it with you to the hardware store so you can be certain to buy the exact size/style necessary. If this is a very old sink, you would be better served to find an old mom and pop plumbing store because they almost always have old fixtures in the back room so your chances are better at being able to find an exact copy.
Great! A lot of ideas in all answers. So, I will try some suggestions before changing all. Very helpful networking
Giguerefr, close the sink , 1/2 cup vinegar, one tablespoon salt, mix in the measureing cup, let the salt dissolve,pore the mix in the sink and let work over night. use old toothbrush to scrub with ,neutralize with baking soda.