Any suggestions for rust in bathroom sinks?

Try peroxide if that fails try some of the toilet bowl cleaners that come with that tear top bottles the works is one brand...also the Mr.clean sponges can really clean up messes toothbrushes can get in around the drain hole and around the faucet riser. When you're done give yourself a treat!

Are they still smooth and have some rust stains or are they bumpy and corroded?

For stains you can use CLR to safely remove rust, but if it’s corroding you need to replace them.

: )

My suggestion is to replace them with new ones.

The chrome is just a coating over steel, brass, or copper. The chrome coating eventually fails and rust begins to show along the edges of the drain. You can clean the rust off but it will come back. Cheaper just to replace.

Try soaking a paper towel with coke and placing it over the drain where the rust is.