When Biegler returns to his house at the start of the film, McCarthy points to the United States Supreme Court reports and asks if they should read "a little Chief Justice Holmes", and Biegler also refers to "Chief Justice Holmes". Oliver Wendell Holmes was an Associate Justice of the U.S. Supreme Court, never Chief Justice. (He was, however, Chief Justice of the Supreme Judicial Court of Massachussetts before being appointed to the U.S. Supreme Court.)
When McCarthy is driving back at night, his car is shown approaching a fork in the road, with a large white sign in the area where the road divides, and taking the left-side (from the driver's viewpoint) fork. The shot immediately shifts to McCarthy in the car, driving, squinting ahead, passing other cars whose horns are heard and headlights seen. The scene then shifts again, back to the very same shot of the car approaching the same fork in the road, even though by then the car would have been well beyond this area, having already passed it several moments earlier.
Several times in the movie one of the characters makes a humorous comment and you hear the courtroom audience burst out laughing; however, when the camera immediately turns to the audience, they are shown completely still and without emotion.
Near the beginning, when Biegler intends to clean the fish he's caught, he lays out newspaper, then places some waxed paper over it. He makes a phone call and when he returns to the kitchen, the newspapers are on top of the foil.
When Paul Biegler talks to Laura in his office, he opens the cigar box, sees that there are no more cigars in it and puts it back on the table. In the next shot, Laura is seen talking. In the following shot, the cigar box is missing from the table.
It is incorrectly regarded as a goof that it would take 2 days to travel from Detroit to Marquette by train. A trip via Chicago, on the New York Central and connecting to the Chicago and Northwestern (which only goes to Ishpeming, not Marquette) would have taken about 1-1/2 days. A short cut could be made taking a Lake Michigan car ferry from Muskegon, Ludington or Frankfort in Michigan to Milwaukee, Manitowoc or Kewaunee in Wisconsin, but connections are a problem; a handy taxi or limo service would be necessary. Going directly from Detroit to Marquette could be done by train in a little over 1 day, via Mackinaw City, but the Chicago & Northwestern would not be used. Many passenger trains on all these routes operated in the 1950's.
Lt. Manion and later Dr. Smith are shown arriving from Detroit on the Chicago and Northwestern Railway. Such a trip would have meant traveling via Chicago and changing trains (and stations). The trip, depending on connection times, would take well over two full days each way.
The Lieutenant is a combat veteran of the Korean War. His ribbons show The Silver and Bronze Star Medals and more. It would seem logical that a veteran with those medals would also have The Combat Infantryman's Badge, which the Lieutenant does not have.