When in the car in 1996 and Cole asks for more music, the switching from station to station is the sound of an analog radio dial (fade out of one station, fade in to the next), whereas the radio being used is a digital direct-tune type. This is likely intentional on the part of the filmmakers as Terry Gilliam frequently places "retro" technology alongside or inside the modern.
When Doctor Railly is spraying the message about the virus on the Animal rights activists place of operations, she clearly is spraying one side of the building, but when they switch to the shot of the graffiti it is on the wrong side.
When James Cole returns from his first trip to the past and is in his cell, he no longer has the bloody wound on his head that he received from his scuffle with the police in 1990. However, when he is being debriefed by the scientists in the next scene, the head wound is back.
When Dr. Leland Goines is tied up in the van and blindfolded with the
biohazard ribbon, the ropes he is tied up with disappear and reappear between shots. Also in the final close-up a magnifying glass appears in his jacket pocket. Also the tape moves from his forehead to his hair and back again in every shot
When Cole runs after Dr. Peters in the airport, he is not wearing sunglasses before or when he is shot, but when Kathryn kneels down beside him, he is briefly seen wearing a pair of dulled, brown glasses.
At the movie theater, Kathryn applies the fake mustache while the couple watch Vertigo. She goes to make the reservations and to change herself. James falls asleep and when he wakes, he is watching The Birds (not listed on the marquis outside the theater). Then he goes to find Kathryn who is in the lobby. When he gets there, only a few minutes after waking up, the music heard is from Vertigo, from the green scene, not from the Birds.
The spider James Cole eats is a Golden Orb Weaver or Banana Spider (Nephila Clavipes). However, that spider inhabits southern regions of the US (mainly Florida), and would never be found in cities in a northern climate like Boston.
Near the beginning of the film, Cole collects a cockroach as it crawls through the snow. Cockroaches aren't active in winter, and wouldn't be found crawling through snow. Also, the cockroach he picks up is a Madagascar hissing cockroach, not found in the US.
The scene in the department store with the tall indoor glass ceilings (shown both in "good" shape and later "ruined"), occurs in the Grand Court at the old Wanamakers store. In later November, the court would have been fully decorated for Christmas, including the traditional light show, which fully takes up one whole wall of the court. This scene was clearly shot beyond the Christmas season.
The television in the sanitarium announces that the featured
picture is the Marx Brothers' Monkey Business, but the clip they show is from their first film, believed to have been removed, and replaced with; The Cocoanuts.
When they are in the car in 1996 and Cole wishes for the volume to be turned up, Kathryn presses the button on the radio. The frequency changes, but the volume goes up.
-- Actually the frequency doesn't change. Initially time was displayed (9:33) and when she presses the button frequency is displayed (103.7).
The scientists of the future should already have had Cole's last message from the airport before they sent him. They played back Dr. Railly's message "before" she had left it in the chronology of the movie, so they could also have had from the beginning all messages that anyone would leave on the secret machine at any time. However, the scientists explain that the process is very time consuming ("one word at a time") and that they had only deciphered Dr. Railly's message a few days ago, so they might not have gotten Cole's message yet.
The bandage on Cole's neck reattaches itself. But the shot where it was re-attached happened sometime later (he had slept some due to the shot) so it's very likely that one of the doctors had re-attached it.
Young James Cole has a scar that runs horizontally almost the entire length. The adult James has no such scar. Some scars acquired as a child may disappear later in life. I had a scar at age 17 and now at 62 is gone. Another scar acquired at age 11 was originally about 3 inches long but is now only about 3/4" long. The section that remains was the most significant part of the injury.
At about 50 minutes into the movie, Cole is caught then shown restrained on a gurney rolling down a hallway. He is struggling and surrounded by orderlies. They stop and Cole begins speaking to Dr. Railly. Cole completely stops struggling but a pony-tailed orderly restraining Cole's feet continues to act as if Cole is still struggling.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
In the final scene where Cole gets shot his fake mustache is seen to be falling off. In the next scene it is fine. The OP thinks this was a goof, however, in the next scene he was laying down. It's very conceivable that the mustache was simply laying there in place since it was still half-attached (it was hanging away when he got shot due to gravity and motion.)