Ahhh... the hat.
The one that you wear year round. In summer, you sweat like crazy inside the hat, making your hair wet or at least oily by the end of the shift.
In the fall & spring, you put the rain cover on, stand in traffic while the water sluices down the hat to the rear... right down the back of your neck. :barf:
In the winter, it keeps heat in, but once you get really bad weather, ice forms from the moist heat.
Like Capn Charlie said, they're the first thing to fall off in a fight or foot pursuit. On days with gusty winds, they can be a distraction as they try to fly away. In the squad car, you have to have a place for it lest you end up sitting on it. All too often you spend money on a spare hat, extra money on dry cleaning a suspect's footprints out of it or replacing it if the bill was trashed during a fight.
The "Mountie" hat (or Smokey bear, DI or Campaign hat) was better in rainy weather as it really kept water off your head. They were also different than what most agencies wore too. But these were also fast to fall off despite the head band in the back and in a wind... well you've seen a frisbee in flight.
I personally like the look a hat imparts to an officer's appearance. The round police hat looks better than the 8-point hat, IMO and the "Mountie" hat can be intimidating (or a reminder of how much you hated your D.I. in the service
).
It used to be that police officers wore ties (clip-on ties for certain) during the cooler months. Nowadays, some agencies permit officers to wear a jumpsuit and baseball cap in the winter which, to me, looks too "Ninja".