We mainly use our "check" card so I said debit. It's through our credit union and is a visa card and there are no fees and probably won't ever be. I use it for just about everything. It's Navy Federal Credit Union. I also only use a cc online.
I think it depends on what I'm buying. If i'm just buying a drink or something under $5, I'll use cash. If i'm buying something at a local business, I'll use cash just because I know the fees they get charged due hurt them. For everything else I use my debit card as a credit card to get the points for traveling.
I try to use my credit card for everything just to rack up "free money" but I also have enough cash on me to cover it. I take the cash as soon as I swipe the card and put it into an envelope in my purse so I know that it goes towards the credit card bill. Once my cash is gone, I'm no longer shopping. It works good for us and we've received well over $500 free in the past year for doing so which helps out Disney vacations