When Martin calls a police buddy to find out who the traffic cop was, he is only on the line for a few seconds. There is no way he could have discovered the cop's full name, where she hangs out and when she's likely to be there.
(around 46 sec) Frasier looks in the rear view mirror and notices there's a police car behind him before the police car actually pulls out behind him, and possibly even before he passes it. Furthermore, given the speed of the passing scenery, there's no way he could have been driving at 62 MPH.
After Frasier blows out the candles on his birthday cake, the candles mysteriously disappear in the next shot when he begins slicing the cake.
When Martin (John Mahoney) calls the police station to get information of the female cop (Maureen Cutler, played by Jane Kaczmarek), he ends the conversations in less than 30 seconds. He then gives Frasier (Kelsey Grammer) a list of information on her that he could not have received in the time he was on the phone.
Just as Daphne and her friends enter the apartment after Martin leaves, the camera starts to pan across the apartment and a second camera briefly moves into the shot on the left.