My door handles had lots of fine scratches in them on the plastic plate behind the handles. On the drivers door, the fine scratches had also got dirt embedded in them, and it all looked pretty messy.
I have managed to completely remove all the scratches, and the dirt, using some Meguiars Scratch-X, normally used for getting fine scratches off paintwork.
Simply squeeze a blob onto one of their foam applicators, and then rub away like crazy on the affected plastic. It should come up a treat unless the scratches are really deep, but fingernail scratches tend not to be.
I wish I'd taken before/after photos now as the difference is amazing
Hope this tip helps.