Rusty marks on bathroom sink top
If the stains are due to rust, any of the commercial cleaners might work if left in place long enough. @dandavis usggested CLR, and IronOut or Barkeepers Friend should also help, if left as a poultice (e.g. covered with a paper towel wet with the cleaner) so that it doesn't evaporate before penetrating. Even "sour salt", citric acid, may work.

If the stain has penetrated deeply, though, it may take multiple applications to make it much lighter.