Champ says Fish is upset about 25 years on the department and no accomplishments. This would have him joining the force in 1950, around the age of 36 given his proximity to mandatory retirement at age 63. In later episodes, he alludes to completing the academy in the late 1930's.
In most large cities, including New York City, arson is investigated by the fire department, not the police. Those cities that do use the police generally have specially-trained officers or units.
As Mr. Cotterman is walking toward Fish to give him some information he requested, the camera tilts down to Cotterman's feet for an instant for no apparent reason then quickly tilts up again.
In the opening scene, when Harris is typing, his fingers don't match the sound of the typing.