The manual for the EM-50 that Harold Ramis' character is reading in the hangar states the manual is from the War Department instead of saying Department of Defense. The Department of War existed from 1789 - 1947.
In the end scene at the airfield, Bill Murray and Harold Ramis' characters (and others) are welcomed as heroes. Ramis's character is speaking to reporters but what he's saying doesn't match up to his lips talking.
When Sgt. Hulka first meets Capt. Stillman, at first Hulka fails to salute the officer. Then after the glare given by Stillman, Hulka raises and lowers his salute without waiting for Stillman's salute. The correct custom and courtesy on saluting is that the enlisted or lower officer rank raises their salute and holds it until the higher rank acknowledges it, usually by returning it but can be done by a simple nod of the head. Then the junior rank may lower theirs.
New recruits are told at the military processing center that they will submit to a urine drug screen upon arrival at boot camp, therefore Judge Rheinhold's character trying to hide his drugs is pointless.
When Russell and Louise are in the General's living room about to play their charade, they move table with a chess set on it that is incorrectly set up. The rule in chess is "Queen on her color, white to the right," yet we can clearly see one of the rooks on a black square at what should be position H1, or the lower right square, which should be white (King's rook for white player), just like position A8 should always be white (Queen's rook for black player). The board was incorrectly rotated one quarter of a turn.
When Ox gets his hair cut at the beginning of basic training, he is shaved nearly bald. When he talks about why he joined the Army, presumably at the end of the same day, his hair looks as if it has grown out for at least two weeks.
P.J. Soles (Stella) begins the movie as a brunette (helping soldiers make it to the bus, driving Murray and Ramis back to barracks after they fight in the tank yard) but becomes a blonde after the mud wrestling arrest (at the general's house getting pancake treatment).
Stella and John pop out of the wood foot locker supposedly in Gen Barnicke's house. An extended cut scene reveals that this scene was actually at their hotel room in Germany as evidenced by the fireplace in the room and John's green Hawaiian shirt.
On the second day of boot camp at reveille and the bugler is playing, the sun is coming up with a few sparse clouds. In the next shot inside the barracks, when Sgt. Hulka wakes up the recruits with the garbage can routine, it is daylight and raining outside (the rain can be seen against the barrack windows).
In the lifted cut scene where John and Stella are having sex in the General's bedroom, which was originally intended as a scene in their hotel room in Germany, you can see a room service food tray in the background with glasses and dishes obviously having been used. An unlikely item in the General's bedroom.
(Extended version). When part of Winger's company is in Lt. Stillman's office, Stillman says they have three days before graduation and that "they won't make it." Yet, later that night on liberty, they go to the mud wrestling event, and after the MPs arrive and take them back to base and Stillman is lecturing them, Stillman says that graduation is "tomorrow."
When the EM-50 is driving up to the resort to pick up their MP girlfriends, the drivers arm is visible wearing a checked shirt. When Russell and Winger get out both are wearing jackets with no checkered shirts.
When they first arrive in Italy, in the background you can see Ox telling Cruiser to get down off the top bunk, because the "Guy on the top bunk, has to make the bottom guy's bunk for him". Then it's repeated when the camera changes to a close up of Ox and Cruiser.
When the mortar shell is fired and ends up knocking down the tower that Sgt. Hulka is standing in after climbing the rope, it sounds like a large artillery shell. The other mortars that had just been fired did not sound like that.
When John, Russell and the MPs hear the distress call in West Germany, the grass is green and trees have leaves. When they cross into Czech territory, the foliage is gone as if it suddenly became winter.
In the scene where the recruits are being measured for their uniforms and Winger apologizes to Sgt Hulka for his earlier wisecrack, the unit crests on the epaulets of Sgt Hulka's field jacket disappear and reappear depending on the camera angle.
Army training, as depicted in the movie, takes nine weeks or so. In one of the first training days the group goes through, the viewing audience can see it is late fall or sometime in winter because light snow is falling and leaves are turning color and many are already on the ground. In just a few short scenes later (which would be no more than a few weeks later in the movie time-frame), Bill Murray's character is attempting to go AWOL alone and the audience can hear crickets chirping in the background, indicating it is summer.
During the rescue toward the end of the movie, when a rocket is fired from the Urban Assault Vehicle toward a guard tower, a crew member is clearly visible crouching at the left of the scene attempting to conceal himself behind part of a building.
In the training shots, where all the soldiers are covered in mud and running through the field with the music playing, in the wide-screen version, the camera crew is visible on the right for a few seconds.
Traveling north from Italy to Germany, as Winger and Ziskey went with the RV to see their girlfriends, and their squad traveled in pursuit, one would have to take a RIGHT into the Czech Republic, and a LEFT into Germany, not a left into the Czech Republic and a right into Germany as the roadsigns falsely indicated.
(at around 18 mins) The 1975 BMW 530i is a US version. You can tell by the big 5 mph crash bumpers and fake plate just resting on the real plate to hide it. It would not be in Germany, a German model would.
While Ox and several other characters gut true boot camp haircuts, taken down to the barest of stubble, Winger, Zisky, Elmo and Psycho among others have haircuts that are too long to be the norm at the start of boot camp.
Every soldier present at the graduation ceremony in a dress uniform is wearing the National Defense Service Medal, awarded to all military personnel who serve during a period where the U.S. is in conflict with another nation; except Captain Stillman. He should have the ribbon for this medal among his awards and he doesn't.
During the graduation review, one of the soldiers is carrying an American flag with gold fringe. American flags with such decorative fringe are meant to be used only indoors; such a flag would never be used in any type of outdoor ceremony.
When Russell points to his medal and says "meritorious service" the medal appears to be a cross. The American medal for "meritorious service" is round with an eagle on it, the medal they are wearing looks British or Canadian.
When John is taking the woman to the airport, he ends up heading north across the 2nd Street bridge into Indiana from Louisville. The airport is actually south of downtown Louisville. However, although many of the vehicles in the city scenes have markings or signs indicating the filming was done in the Louisville area, nothing in the dialogue specifies that the setting is Louisville or any other specific city.
When the men assemble on the parade ground for their impromptu performance at graduation, a yellow hash mark is visible on the ground that tells them where to turn right as they double-time into position.
There is a scene where Cpt. Stillman (John Larroquette) pulls the pin out of a live grenade out of frustration that there was "no action". However, if you look closely at the grenade, it is painted blue. In the military, blue is the code color for "Inert" or disabled. This is to make sure that live explosives or weapons are not used in training sessions for obvious reasons.
The weight of the duffel bag Bill Murray was carrying on his back while marching presumably to their assigned barracks. Initial issues of military uniforms and accessories will weigh a lot heavier than what they were carrying.
The goof item below may give away important plot points.
When rescuing the rest of the team from the prison holding cell, Winger and Stella attack and disable the 3 guards outside the locked room. Since these guards have a prisoner (Cruiser) out of the room interrogating him, it seems reasonable that at least one of these guards has keys to that room. With keys handy, there would be no need to "blow the door" with plastic explosives, or have Ox knock it down.