Any way to get tobacco stains off my builder-grade bathroom sink?

https://www.hunker.com/13424083/how-to-remove-cigarette-stains-from-a-bathroom-sink

Magic eraser and elbow grease, depending on the material

Are they stains or burn marks? If stained, I like the Dollar Tree Awesome Orange, seems to melt nicotine off walls, ceilings and cabinets.

I adore Krud Kutter for stubborn stains. You may need to use a variety of products to find the "right" one. I personally would begin with Barkeepers Helper, a non-scratch cleanser.

Try Cerama Bryte, which is made for cleaning glass stovetops, which often have burned-on stains. It is a liquid/paste you apply to area to be cleaned, let sit a few minutes for bad stains, then scrub and rinse off with a wet paper towel. Cerama Bryte is sold at Home Depot, Lowes, etc., often near where stoves are sold.

Try Chlorox spray on cleanser with bleach or 409 with bleach.