I used my visa debt card in Cozumel, Jamaca, and at Grand Caymen. I did stick to the main areas though, as to avoid language barrier problems. In all the main areas they took visa and mastercard. I did notice that a few places did NOT take Discover or American Express. Visa and Mastercard are still the standard, so in my opinion they would be the best to use.
I've pretty much used them all and haven't had a problem. I would hesitate to use a debit card, just in case there is any problem, the money would already be taken from your account. Use the other major cards and then that if needed.
I agree with the above post about not taking a debit card. The only reason I used mine was that I had a separate account for the cruise, a trick I picked from my dad. That way I knew exactly what was coming out, and nothing was linked to my direct deposits.
I also agree with those above. I did notice that Discover was not as popular. One thing as a warning to all. If you are going to use a card in the islands, call your credit card company before you leave. Some companies have a red flag if the card is used outside the US if they are not warned first. I was behind a lady in Cozumel who had not used her card in months, and it had been cut off for lack of use.