My inherited, plastic, many bathroom sink has been the bane of my cleaning life (I don't do much but like a clean bathroom).
I don't usually have bleach in the house but I tried bleaching and various other 'mainstream' products. My local Oxfam have started doing their bathroom cleaner and as I use mostly their products I used it>
It cleaned the basin 
. After 2.4 years in this house it is white.