Last Of The Mohicans has the worst and most numerous mistakes I’ve ever seen in a movie. I noticed many of them on my own at first viewing, saw many others after reading the following:
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Detroit Rock City
The guys are in a car, being chased by the Guidos. There are close-ups of the car, then medium range shots, alternating back and forth.
Is the driver’s side window up or down? Or both?
Nobody’s Fool
Sully (Paul Newman) is having a conversation with Toby (Melanie Griffith), in Toby’s kitchen. As Sully prepares to leave, he pulls on one glove, then the other (it’s winter). The camera shot cuts from Sully to Toby as she says something, then cuts back to Sully. As he says his next line, he pulls on one glove, then the other.
(This is one which I actually noticed while watching the movie in the theater. 
)
In Transporter 2 our hero is being pursued in his car, an Audi A8 W12. However, the W12 logo on the front grill keeps appearing and disappearing throughout the sequence.
Nope, but at least two of the X-Men can.
Not a movie, but watching Game of Thrones this week I was strongly bothered by the opening scene of people running in the snow and not being able to see their breath.
One of my favorites is in The Man With The Golden Gun, where Bond is fighting a couple of heavies, and one of them bumpb a mirror. In the mirror you can then see a couple of crew members, a light, some chairs, etc.
About 15 seconds here.
Another favorite is in Commando where Arnold chases Sully in his 60’s Porsche 911, which turns into a 70’s Porsche with a huge whale tail on it, then back to the 60’s one. After Arnold flips it over, he pushes it back onto its wheels and drives away in it, where you can see there’s not a scratch on it.
I used to spot boom mikes all the time in B-grade movies. Not so much anymore.
No specific movie to cite, just almost every “jungle movie” from the B&W days, Tarzan especially.
One camera angle has the natives chasing a beast (lion let’s say) through dense forest/jungle/bushes and the next camera angle has the same critter running in an open plain with maybe one or two trees. It’s especially bothersome if the action is for bwana to shoot at said critter. When he aims, the jungle is behind him and when we see the result of the shot the critter is out there without a tree in sight.
Never fails.
I always cringe when they dub in lion roars when a lion is pursuing prey (even worse when they do it for non-lion cats). Cats are silent when on the chase, conserving oxygen for the exertion involved. Lions usually only roar when looking for mates or establishing territory.
Of course the greatest one of all time was the MST3K movie Space Mutiny, with the woman who was shot and killed showing up in the next scene. “Hey - she’s dead!”
Space Mutiny. It was special, yeah.
“Alright, look alive, people! Oh! Sorry, Debbie.”
Not necessarily a mistake. In extremely cold and dry climates like Antarctica you can’t see your breath most of the time. There’s no moisture or pollution in the air for the droplets from your lungs to condense around. I think both those conditions would hold true in Westeros.
One of my favourites, partly because Lucas enhanced it with a ‘thump’ sound effect for the 2004 DVD release.
Here’s one that’s bugged the hell out of me for over thirty years:
Star Wars IV, the scene where Tarkin and Vader learn that Leia lied to them about the rebel base. Vader says, “I told you she would never consciously betray the rebellion”, and then shakes his finger awkwardly at Tarkin, obviously a case of the voiceover being out of synch. Pissed me off all over again when it wasn’t fixed in the remaster.
Same thing when Luke and Threepio are in the Speeder looking for Artoo. Threepio is sitting there moving his fingers like he’s telling a story.