K tells J, "500 years ago, everybody 'knew' the Earth was flat." The Earth's spherical nature was something that most scholars and all navigators were aware of as early as 400 BCE and has been common knowledge for the past 1000 years.
When the bug (in the Edgar suit) enters the morgue he is carrying a shotgun and places it next to the window before he rings the bell. In the next scene he is attacking Laurel with the hand gun he took from the tow truck driver with no shotgun in sight. When J goes to the window to ring the bell the Shot gun should be leaning against the wall but is nowhere to be seen.
When the Bug goes into the farm house wearing Edgar's skin he drinks sugar water, and then Beatrice tells him that his skin is hanging off his bones. He looks in the mirror then grabs his hair and pulls his skin tightly to the back of his head. Beatrice faints and the next shot shows the bug still pulling his hair back, but the skin is no longer stretched.
When Edgar breaks into the store you can see the Zap'Em truck outside. The roof is smooth and undamaged while in the earlier scene when he drives into the city the roof is torn up and his ship is sticking through the hole in the roof.
Kay says that when the replacement to CDs comes out, he'll "have to buy the White Album again." Later in the movie, we see that Kay still uses 8-track tapes, which means he likely hasn't gotten a new White Album in the course of a number of format changes (tapes and CDs).
The first time J and K are in the morgue with the coroner, J and the doctor leave to examine the next body and K starts to examine the first body. When the camera angle changes we can clearly see body #1 behind them but K is nowhere to be seen.
When Mets player Bernard Gilkey is hit on the head with the ball, he is wearing jersey #23 (Gilkey's number), and the jersey has the standard "Mets" script across the chest. Later, when he is seen on the cover of the tabloid, he is wearing #20, with the 1993-94 script that has a "swoosh-tail" attached to the "s" underlining the word "Mets."
When J is chasing the first alien in the movie, about 10 & half minutes in, J gets inside a yellow work truck to catch up to the alien, just after the close up of J inside there is an outside/distance shoot and a stand in is visible, visible because he is white and has more hair than J.
In the beginning of the movie, when "J" chases the alien into the building, he is initially seen running clockwise up the circular ramp, but after the scene cuts away and comes back, he is running counterclockwise.
When Redgick is stopped with his pregnant wife in New Jersey, the World's Fair-towers in Queens can be seen briefly in the background. Since the World's Fair took place in Queens, it's impossible to see them from New Jersey.
In the end where we see Manhattan's place in the galaxy, the camera zooms from a location near the center of the Milky Way. According to most scientists, earth is actually located on one of the galaxy's "arms".
As J chases the alien at the beginning of the movie, the
chase starts at Grand Central Terminal. In the next shot, the alien is seen running downtown (the park is on his right) to get to the Guggenheim Museum. The Guggenheim is uptown from Grand Central, and so is Central Park.
When Agent K is looking up his lost love, he types "Truro",
then zooms in on Cape Cod. As he closes in, he misses Truro on the map by about 50 miles' drive (the town/village he zooms to is actually closer to Forestdale or Wakeby). Also, the suburban grass lawn is atypical for Truro. Homes are mostly surrounded by pine needles, brush and dunes.
When looking at the star chart of Orion's Belt, Alnitak can be seen labelled as the middle star. From the northern hemisphere, Alnitak is the leftmost of the three stars of the belt (Alnilam is the middle star, Mintaka is the right hand star).
As Zed explains Orion's Belt to J at HQ, he rotates the constellation Orion vertically on a screen display and zooms in, showing the Belt's stars (Alnitak, Alnilam and Mintaka) to be in a roughly straight line and of equal brightness. In reality such an "overhead" projection of the Belt would show a long triangle with a very narrow base (Alnilam is about twice as far from Earth as the other two stars), with Mintaka on the right nine times brighter than Alnitak on the left.
Kay says the MIB agency fund themselves through "a few patents", one of them being the microwave oven. However the microwave oven was invented in 1946 and is sold commercially since 1955 - before MIB even existed as an agency - making it impossible for them to patent it. Raytheon, established in 1922, owns the patent.
J and K exit the morgue at night and go directly back to MIB headquarters. J goes to his computer and scans for his lost love in Cape Cod, which is in the same time zone as NY. When the image is located, it appears to be daylight when it should be in the evening.
(at around 35 mins) Agent K says that an alien traveled 17 trillion miles. That distance is equal to about 2.89 light years. But Proxima Centauri, the sun's nearest stellar neighbor, is 4.22 light years away. So where could the alien have come from?
When Edgar drinks the sugar water you can see that most, if not all, of the sugar stays in the glass (sticks to the glass). He didn't give it time to dissolve. Therefore, he barely gets any sugar with the drink.
Kay unplugs the camera in the interrogation room but that would only stop the camera from recording from that point on. It would have recorded Kay walking into the room and up to the camera which defeats the purpose of unplugging the camera to prevent evidence of Kay being in the room and taking Jay out of the room.
When the diner owner asks Edgar the bug "what happened to Ivan?" he replies, "(I) gave him a break". But the French subtitle reads, "Laisse le tranquille" which means "leave him alone" which could also be worded "GIVE (present tense) him a break" as an imperative.
When riding in the tunnel, J puts on an 8-track and listens to Elvis. J quips, "You do know Elvis is dead?" to which K replies, "No he isn't, he just went home," implying he is an alien. Earlier K reveals the aliens first made contact in 1961 - five years after Elvis Presley made his breakthrough performance on The Ed Sullivan Show. However, 1961 is only when they made contact, not necessarily when they first landed.
In the opening scene, which is set in either California, Arizona, New Mexico, or Texas, we see New York license plates on the cars. However, it is possible that the MIB drove from New York to one of those states as the guy was bringing immigrants over the border.
In the tunnel, J falls upside down and appears to have a Caucasian leg. However, the behind-the-scenes featurette shows that Will Smith performed this scene without a stunt double, so this must be a trick of the light.
When James chases the alien in his first scene, after he stops him, he sarcastically explains to him what NYPD stands for, "Nock Your Punkass Down". The word "knock" starts with a "k". Obviously the statement is sarcastic.
In the tunnel when they're upside down, based on Kay's hand in relation to his seat (right hand holding tape and entering scene from right side places car upside down in shot), he places the Elvis Presley eight track tape into the player right side up, meaning the player is installed in the car upside down. The player also says "Eight track stereo" on front cassette door right side up to the shot, meaning upside down in the car.
In the beginning of the film when Edwards is hanging off the back of a van chasing the guy, as the van turns into a street you can see nobody is in the driver seat. There is a person sitting in the passenger seat looking back through the window the whole time.
When Zed, Kay, and Jay are trying to determine where the bug will go to leave Earth, Zed gives Kay possibilities to look up on his computer while Jay is in the foreground looking at the picture of the UFOs from the World's Fair. In the background, you can clearly see Kay pretending to type in Zed's suggestions on his keyboard.
As J is being shown the screen where they constantly observe the aliens, the clips on the screen are being endlessly looped. A screen close to the center, for instance, shows a man getting comfortable on a couch over and over again.
After Edgar kills the Arquillians in the restaurant and rushes outside, we follow in a through-the-window shot as Edgar tosses a bystander off the sidewalk and then menaces a second in a close pass. But in the immediate exterior shot on Edgar from the sidewalk, the second bystander has vanished.
When the Edgar alien crashes out the morgue window, carrying Laurel, white shorts can be seen sticking out from between Laurel's brown shorts, and her pantyhose. These do not appear again, despite numerous tumbles by Laurel that should have revealed them again.
The MiB get splattered by the bug's goo at the end, yet in the next scene (just before K asks J to flash his memory) they are wearing clean suits while Laurel Weaver still wears dirty clothes. In the DVD commentary, Barry Sonnenfeld insists the MIB suits are made of a "slime-repellent" material.