I don't know what all you've tried so I'll just list what I would do.....
1. Magic Eraser (haven't found an off brand that works as well)
2. Magic Eraser and some Soft Scrub
3. CLR
and if none of these work and we are decided that the sink couldn't look any worse.......
4. CAREFULLY use gloves and eye protection and you might try The Works toilet bowl cleaner...put a little in a throw away plastic container (less than an ounce) and take a cotton swab and dip it in the cleaner and apply to the stains. It may leave a light blue stain but it will dissipate w/use.
Good Luck!