Broken Steel 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.
Broken Steel 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.
Broken Steel 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.
Point Lookout (add-on) A police hat can also be found on top of a television in the Grower's shack.
It seems the police officers before the Great War used .32 pistols or police pistols, as their remains are most often found near one.
The adult version of the police hat is contained in Fallout 3's game files but is not actually used in the game. However, Knock Knock and Knick Knack (located in Little Lamplight) both wear the child version. The only way to obtain the adult version is through console commands or the add-ons.