In the Belfort offices there is an electronic stock ticker. It showed twice during the movie a stock symbol called HPQ which is Hewlett-Packard's symbol since they purchased Compaq Computer, sometime after this film's time period.
London's tallest building, The Shard, can be seen in the background of the scene between Jordan and Aunt Emma in Kensington Gardens, despite the scene being set a couple of decades before it was built.
When the marching band comes into the office, the harnesses that the percussionists wear to carry their drums are visible. These are modern harnesses, designed with a strap going around the players' backs to offer additional stability to the instrument. This style of harness did not exist during the time frame of this scene.
When Jordan tells the agent that he "got the real stuff, Lehman brothers, collateralized debts obligations etc" he could not have known this info at this time because at the time frame of the movie CDOs were extremely obscure and not very widely known or traded on a wide scale by major firms like later on in the late 2000s.
When Jordan and Donny talk about how Donny married his first cousin, Donny claims that there is a 60-65% chance of birth defects stemming from his first-cousin marriage. In fact, even with first cousins, the chance of birth defects occurring from an isolated cousin marriage is not significantly higher than average - an estimated 3.5-4% as opposed to 3%. Inbreeding can be a bigger problem if close relatives continue to marry each other over several generations (which is why the Spanish Habsburgs went extinct, but only after several generations of serious inbreeding, or why Pakistan - where cousin marriages are very prevalent - has a particularly serious inbreeding problem), but even in these cases the chance varies wildly from 10-40% depending on other factors.
In the opening narration when Jordan points out that his Ferrari was white, not red, the car he is driving is a Ferrari 512TR, the refreshed version of the Testarossa introduced in 1992. Later on when he and Naomi pull up to Naomi's apartment, they are in a white ~1986 original Ferrari Testarossa.
When Jordan is telling a client about Aerotyne International on phone, Dwayne turns back on his chair to see him and two men are standing by him. In the very next shot Dwayne can be seen in background sitting while facing his table and those two men disappear.
When Donnie and Jordan are fighting each other under the influence of Lemmon 714, Jordan gets tangled up by a telephone cable. One cut later, the cable disappears and Jordan is able to move freely again.
Jordan and Donny are sitting in a bar and talking. At the moment Donny says: "I'm not gonna let someone else F**k my cousin", the cigarette he is holding is almost burned up, one shot later it's only half smoked up.
After he has lost his first job on Wall St, Belfort arrives at a dinky plaza in Long Island. He parks his car at the closest parking spot, right in front of a "brokerage" to apply for a job. In the next two shots you can not see the car anywhere in the parking lot, as he passes through the door, and then we look out the window of the brokerage. In both shots we see the parking lot clearly but his car is not in the lot. The car was parked no more than four, or five feet away from the entrance.
Shortly before Naomi pours a glass of water into Jordan's face for the second time we see him kneeling in bed reaching out for her with both hands (camera behind Naomi). After the cut (camera from the side) he sits on the bed with hands on lap.
During the introduction of Steve Madden, Jordan is rallying the stock brokers and puts his hand around the neck of one stock broker. Continuing his speech he puts his around the neck of the next stock broker. The camera goes to a front shot of Jordan and he is in the process of putting his hand around the neck of the same stock broker.
When Jordan asks Chester to sell him the pen, he extends the pen in his left hand. However in the next shot it is in his right. In the next shot it is back in his left before he switches hands to hand the pen to Brad.
When Jordan Belfort and Donnie Azoff decide to start their own brokerage trading company, there is an exterior shot showing them stopping at a small, abandoned auto garage for sale to see if it might serve as their firm's first location. The next scene is an interior shot of the garage showing them setting up for business. However, the interior size of the building shown is of a much larger building than what was filmed in the exterior shot.
In the flashback scene of Jordan leaving the Country Club while under the influence of Ludes, it's listed that he's driving his Ferrari. He's not in his Ferrari, he's in an early 80s Lambourghini Countach that is totally different than anything made by Enzo and the crew.
At the beginning of the movie, when Jordan Belfort reveals that on a daily basis he takes enough drugs to supply Manhattan, Long Island, and Queens for a month, he says that he takes Adderall to stay focused. Adderall was not released until 1996, and was not released in a generic form to the market until 2002. Given that the film spans from 1987 to sometime in the 1990's, using "prescription Amphetamine pills to focus" would be much more factually accurate and understood than using "Adderall to focus" during the 1987-1990's time span of the film.
When the group is getting ready to smuggle the money to Switzerland, Jordan claims that Brad's wife has a swiss passport due to having been born in Switzerland. However, birth does not grant a child swiss citizenship, at least one parent has to be of swiss nationality.
Towards the end of the film, Jordan puts his daughter in a late model R129 Mercedes SL500 to take her away from Naomi (either a 1999 or a 2000). Not only does this car not fit the time-frame of this film, but the way he engages the parking brake in reverse (with his right hand pulling upwards) is incorrect on this type of vehicle. The parking brake is located near the brake pedal and must be pressed down with your left foot.
When Jordan is having a flashback to driving his Ferrari out of the country club while high on Lemmons, we see him yank the steering wheel to the left; but in the following shot, the car moves forward and turns to the right. However, this is not a goof since Martin Scorsese has confirmed that where Jordan has taken drugs the editing becomes odd on purpose, with continuity issues to make events flow oddly.
The goof items below may give away important plot points.
When Jordan Belfort is arrested, the police vehicles are "current" models. The Old Brookville police interceptors are the new, current color scheme for the department. In addition, they are equipped with LED emergency lights, not yet introduced at the time the of the events.
When Jordan and Naomi have sex for the last time (before she declares she wants a divorce), Jordan is seen with his trousers off in one scene. But in the next scene, when Naomi walks away from the bed, Jordan is seen lying in bed with this trousers on.
Towards the end of the film in the court room scene Agent Denham and his party are all standing at the prosecutor stand. As the shot changes and Jordan is led away Agent Denham and his party are seated.
While Jordan is being arrested at his home after recklessly driving his Lamborghini Countach home from the country club pay phone, the officers are driving 1998-2011 model Ford Crown Victoria Police Interceptors, although the scene is set in ~1990.
When Naomi is taunting Jordan by showing she's not wearing underwear during her Lysistrata-style sex strike, and he tells her she's being filmed, safely modest white panties can be seen just before she holds her legs together.