In a passenger car, they are totally unnecessary and cause road salt and other corrosive elements to accumulate in an area that is already very vulnerable to rusting.
Personally, I think they (like bras) look silly on most cars and could cause more damage than they prevent if not installed correctly. I think it makes more sense to just have stone chips repaired if/when they occur. As a fashion accessory, mud flaps are in the same category as curb feelers.