Right now I am considering CapitalOne NoHassle, but I've seen a lot of companies using various AmEx cards.
For example I remember when I bought the first iPhone for $600, after Apple reduced the price by $200 and refunded $100 with a gift card, Amex refunded (even if the iPhone wasn't covered by their policy) everyone by the $100 to make up for the difference.
Amex has credit cards, but they are mostly (IME) associated with charge card accounts.
"A charge card is a plastic card that provides an alternative payment to cash when making purchases in which the issuer and the cardholder enter into an agreement that the debt incurred on the charge account will be paid in full and by due date (usually every thirty days) or be subject to severe late fees and restrictions on card use"
So, you tend to optimize for different things (features, rebates, discounts vs. Interest rates and payback optiins (generally))
A credit card allows you to float debt, and therefore can improve cash flow at the expense of interest.
Charge card: you pay an annual membership fee and then if you pay off the balance in full each month then there is nothing more to pay.
Credit card: you pay at least the minimum amount when the statement comes and you get charged interest on the balance. There may also be transaction fees for cash advances, etc.
Both the annual fee and the interest rates are highly variable so it pays to shop around. Personally, I think charge cards cost less in the long run.
Another option is a Debit Card, which uses funds in an account. There are some account charges. From a fraud resistance point of view, as long as you don't have much money in the linked account, the debit card poses the lowest financial risk, but can be a hassle to keep transferring money between accounts.