This first time Murphy (Burt Reynolds) visits his apartment, he switches on his 1930s era radio, which instantly begins playing music, as a modern one would. This is incorrect - a radio of that era contained vacuum tubes, and it would have taken the radio several minutes to "warm up" before any music could be heard.
In the scene where Lt. Speer breaks into the mobster's home to get the slug upon which to run ballistics the mobster asks Speer if Speer has a warrant. The movie takes place prior to 1934 and the repeal of prohibition. There was no requirement for police to have warrants to search citizens' property or to seize such property until the 1940's.
In the gunfight scene in the street, when Lt. Speer is walking down the street with the pump-action shotgun, the audio is of the action being operated in one fluid motion, while the video shows the action being pulled open, Speer pauses momentarily, and then closes the action, with no sound.
In the scene where Spear gets into a shoot-out with the thug in the hallway and stairwell, he shoots the thug a total of eight times, all in the upper chest area. However, later when he and Murphy are examining the bodies in the morgue, there is an overhead shot of the same thug and he has no upper chest wounds of any kind.
The doctor tells Lt. Speer that Ginny broke her arm when she was hit by the car. However, when she reappears at the end of the movie singing in the club, she is not wearing a cast or showing any sign of injury.