As Frank Serpico is driving across the bridge in his Studebaker Lark to report for his first posting as a patrolman in 1960, many of the cars in the background are models from the mid-1960's to early 1970's. (This is true in many of the early street scenes as well.)
In the opening sequence of the movie that shows Serpico having been shot, it is raining heavily while he is being rushed to the hospital in the police cruiser. Towards the end of the movie where this sequence of events picks up again, it is not raining.
As the cops are monitoring the drug dealer's apartment near the end of the movie, there is a French Renault Dauphine parked in the front of the building. The Dauphine disappears after Frank enters the building. The gray car parked across the sidewalk on the other side of the street disappears as well, once the police bust the 2 users leaving the building.
Serpico was shot in his face and operated on. He subsequently wakes up after the operation with his beard intact. The hospital would have shaved it to cure the injuries caused by a bullet trespassing his cheek.
During the shooting gallery scene, Frank and a few other cops are practicing their marksmanship. All the bays are full except for bay two, which has the shooters table up, to allow access to the gallery itself. In the reverse scene, the table is down.