Flat tires are a PITA because they're usually NOT covered by the rental agencies' collision damage coverage -- or your own credit card coverage.
In the UK once, I got upgraded to a Mercedes -- which was great until one of its tires got a nail puncture. It cost me over $200 to replace the tire. To my amazement, AMEX paid the tab (less the VAT) from their car insurance program. When I read the policy, I was pretty certain that it said the coverage didn't include tires, but they paid, so I didn't question it.
