At the beginning of the movie, Bud Fox and Marvin say Gordon Gekko was shorting NASA stock right after the Challenger explosion. The scene is set in 1985, but the Space Shuttle Challenger exploded January 28, 1986.
Early in the movie when Bud wakes up in his apartment and turns on his computer he see's it is Gordon Gekko's birthday. The date on his computer shows that it is May 6 1985. Later that same day after his meeting with Gekko he is back at the office when Marvin offers to take him to a New York Knicks game that night. In 1985 the Knicks last regular season game was on April 13th, nearly a month earlier. The Knicks did not make the playoffs that year, therefore there would have been no Knicks game on the evening of May 6, 1985.
Gekko says his 3-year-old son "had the highest IQ score in his class." No known IQ test is given to 3-year-olds. Obviously, Gekko - as is his wont - is boasting about his son without any real basis for the boast.
In his first meeting with Gekko, Bud says Bluestar has "80 medium body jets." Aircraft are either narrow body or wide body. Bud may be revealing his ignorance about aircraft, though he seems fairly well-informed about Bluestar and the airline business.
When Bud first enters the party at Gordon's house in the Hamptons, Gordon introduces him to the party guests, starting with Sam Ruspoli, Darien's boyfriend. Later, after talking with Darien, Kate Gekko again introduces Sam to Bud, by asking "if they have met." Sam replies "no", even though he met Bud less than 5 minutes earlier.
Various characters make a hash of the description of the occurrence that gives Bud his entree to Gekko, that being the not-yet-public news about Bluestar that Bud passes on to Gekko and Gekko trades on. According to Bud's father it was a ruling by the FAA in the course of the investigation of an accident (Bud later calls it a crash). The FAA doesn't investigate airplane crashes: the NTSB does. It's an independent agency, not part of the FAA. Also, when he's talking to Gekko, Bud says it's a lawsuit, not an investigation, and refers to plaintiffs. Neither the FAA nor the NTSB makes rulings in lawsuits: judges do. Finally, both Bud and Gekko clearly imply that Gekko trading on the tip from Bud is wildly illegal. Under the Supreme Court's Dirks decision (1983), Gekko's trading would appear to be legal, as Bud's dad had no intention (or even a thought) of gaining some benefit by mentioning the ruling to Bud.
(at 43:27 and 1:48:14) In the two instances that Bud calls the Wall Street Chronicle to tip them off about the shares, although the occasions are days apart, the same people are sitting/standing/walking at exactly the same places with the same dresses (down to the wrinkles of the rolled sleeves of the manager), with the same set of pictures scattered on the table. Obviously, the two scenes were shot one right after the other.
As Bud is leaving Gekko's office during their initial meeting, Bud's tie is perfectly straight and all the way up to his neck. When Bud reaches the other side of the door, his tie is crooked and half way down his chest.
When Gordon Gekko walks away from Bud Fox in the dressing room, Bud says, "I am not just another broker..." When Gordon comes around the lockers he is wearing a suit with a pair of sneakers (full screen).
When Bud Fox and Gordon Gekko are in the park talking after the Bluestar Airlines deal, Gekko's coat is first seen soaked in large areas by rain, then changes slightly less, then almost completely dry, then back to the original condition.
At Bud's apartment, when his father shakes Gekko's lawyer's hand as the lawyer is about to leave, the editing on the handshake has one complete handshake (extend, shake, pull back) on Carl's line then just the lawyer pulling his hand back from the handshake on his line. The lawyer appears to be pulling back twice from a single handshake.
When Gordon Gekko is sitting on the couch watching the financial news and the plummet of Blue Star, his son Rudy moves in front of the television in a toy car, from right to left. When the camera pans out to show the whole room, Rudy is off to the right of the television, with a water bottle in his hand.
When Gekko is on his phone talking to Bud by his Hamptons beach house, he refers to the beauty of the sun coming up. The sun appears to be rising behind him, to the west. Beaches in the Hamptons face south, so the sun should be in front of him, off-camera.
Bud readily admits his guilt and remorse about using inside information, as evidenced by his tears in the office when they took him away, and his willingness to cooperate with the SEC, the police department and the postal inspector. Since it was his first offense, and he paid back the illegally obtained funds, he should have received probation, not jail time.
Marv tells Buddy about Gekko that "15 minutes after the Challenger blew up" that he was profiting off NASA's stock. While NASA is a government agency and thus has no stock to short, the notion that Gordon Gekko would do such a thing is clearly intended by Marvin as a joke. It is hyperbole on the theme that Gekko always thinks of a way to make a profit before anything else.