In NYC, rental car companies will not accept debit cards for a car rental "hold", though one can pay final charges that way. A few years ago, one could use debit cards as credit cards for car rentals, something to do with how they were processed/coded, but that was changed and now the system will not accept debit cards, even in credit card mode.
There are so many horrible stories of people's bank accounts being cleared out by debit card fraud. Every thing from email fraud, where one is directed by a supposed email from one's bank to click on a link and enter your pin/password because of a "problem", to scanners placed at ATM machines to read pin numbers and collect debit card data.
The other thing one must watch out for is anytime one's card is taken out of sight to complete a transaction. Things such as at a restaurant, or store where the clerk has to "go to another register". Too many script kiddies out there with portable scanners. If you use your debit card and they get your PIN, you could be cleaned out before arriving home.
L.