I can't imagine any person using sharpie on their face. For one it takes forever to remove and second it's not good for you. I watched an interview of her's a couple of months ago and she stated that she uses the liquid eyeliner from Cover Girl.
i think i remember her saying in an interview that she was on a plane and she didn't have eyeliner so she went to the bathroom and put sharpie on her eyes :)
Honestly I don't know. But it doesn't really matter. People go all crazy on celebs and how they look and their makeup and personal life. If I were famous I wouldn't even want to be. You will be criticized about everything. Whether its about who you're dating or what you wear. So its normal to hear rumors like that, but its just that you shouldn't even care. She can sing. That's all that should matter in a career but people stress over other careless things. So if I were you I would forget about it.