Police hat

The police hat is a piece of headwear found in the Fallout 3 add-ons Broken Steel, Point Lookout and Mothership Zeta, and in Fallout: New Vegas.

Fallout 3

 * In the presidential metro, After you exit the hall with a few flags and the statues of presidents immediately turn right, it is on the bench there.
 * By a skeleton on a bench near some of the tracks. To find this is easy, after you exit M.A.R.Go.T.'s control room you will see an American Flag, go that way and turn left towards the tracks, while facing the tracks make another left. Go towards the caved in part of the tracks, and near the wall you will see a skeleton with a .32 pistol and a police hat.
 * One is also located on the floor in the office with the safe and two cash registers in the Old Olney S. Wilson Building.
 * A second one is located in the center of the Old Olney S. Wilson Building. It is accessed by jumping from a ledge adjacent to the fallen GNR billboard onto the empty deathclaw cage. The police hat is next to a skeleton on a small platform at eye level.
 * A police hat can also be found on top of a television in the Grower's Shack.
 * One is located inside the lighthouse halfway up the stairs in Point Lookout.
 * There is a police hat just inside of the People's Bank of Point Lookout next to the head of a skeleton.
 * One can also be found in the add-on Mothership Zeta in the cryo lab among a wide variety of junk.

Fallout: New Vegas

 * In Westside West Entrance, Dermot is wearing one.
 * Some can be found in the Sierra Madre.

Repair
Curiously the police hat can be repaired by the Enclave officer uniform, among others, allowing you to repair it easily with Enclave respawns when you run out of police hats. Among others is the merc grunt outfit, Enclave officer uniform, pre-War hat, pre-War bonnet, (dirty) pre-War casualwear, and (dirty) pre-War spring outfit.