Kenny uses the expressions "cool" and "blows my mind" during the bar conversation. "Cool" was an adjective, not an exclamation, in the early 1960s. It and "mind-blowing" were usages that came in with the Hippy era of the late 1960s.
The aria "Ebben Ne andro lontana" from Catalani's opera La Wally is used in the film. The recording was released in 1992 by Naxos. The singer Miriam Gauci was born in 1957. That would have made her 5 years old at the time of the movie.
There are several typographic anachronisms including the use of Hoefler & Frere-Jones' Gotham (2000) for all of the office door signs at the university (also used in the movie's titles), and Trajan (1989) and Gill Sans (not popularized in the US until the 1970s) in the bank's logo.
Jim's rolled car is a 1963 Rambler Classic station wagon. This model is clearly identifiable as a 1963 model as it had a different grille and tail lights than the 1962 model. The 1963 model would not have been available at the time of the crash which would have been in April of 1962.
When George is leaving for work in his car and he is looking at the girl who lives in the house across the street dancing in slow-motion, the house behind her has dimensional-style shingles which were not available until the 1980s.
The dancing ashtray: When the main character is in front of his class, the ashtray on the desk keeps moving from back to front depending on the camera angle. Obviously they shot the scene from one angle, went back and did it at another angle and in the meantime the ashtray had been moved. The intercutting of the angles resulted in the "dancing" ashtray.
When George is peeking at his neighbors from the toilet, his neighbors house is on the right side of his house. But when he's leaving for work, he drives from the right side of his neighbors house, which suggests his neighbors are on left side of his house.
In the scene nearing the end of the movie, where Falcon goes to the open door to view the full moon, there is an owl in the scene. This owl in the movie is a Eurasian Eagle Owl, only found deep in the forests of Europe and Asia. There would never be a wild Eurasian Eagle owl in L.A.
The white Corvette in the parking lot is indeed a 1963 Stingray. The split rear window was only available in 1963. Production started in 1962 for the '63 model year and is possible that it could be an early delivery.
The goof item below may give away important plot points.
In the final scene of this superb film, while sitting on the bed and beginning to have his heart attack Colin Firth lays his left hand on the right side of his chest then grabs his right arm. While this is far less common than the pain being on the victim's left, it's possible. However, in subsequent shots, when he's fallen to the floor, he's holding his left arm with his right hand.