Continuity mistake: At the beginning when Indy is talking to Spalko, his hands keep alternating from being in his pockets to just resting at his sides between shots.
Visible crew/equipment: During the jungle chase, Marion jumps out of the truck, leaving Dovchenko driverless. A few shots later, Spalko fires on the two amphibious vehicles, and when Mac removes his cap to check his head, the actor's modern wired device (earbud or mic perhaps) drops out from under his scarf/shirt and hangs over his shirt. Later, in the chamber with the crystal skeletons, Mac admits to lying about the CIA, then as Spalko climbs the chamber stairs, it looks like her lav mic is a bit visible between two upper buttons. (01:17:10 - 01:44:15)
Continuity mistake: At the start, just as the carful of teenagers joins the convoy of military vehicles on the road, there are very long shadows being cast on the road, which vanish in the following shots. Their long shadows return when the convoy turns onto another road, but disappear again as they arrive at the main gate. (00:00:45)
Trivia: In the Mayan temple, Indy comments "I have a bad feeling about this." This line was a running gag in George Lucas' Star Wars saga, and was used by Harrison Ford at least once. (01:46:10)
Trivia: When in the tent and forced to face the Crystal Skull, Indy dismisses the skulls as being from "Saucer Men From Mars". Indiana Jones and the Saucermen from Mars was actually an early title for the film.
More trivia for Indiana Jones and the Kingdom of the Crystal Skull
Dean Charles Stanforth: We seem to have reached the age where life stops giving us things and starts taking them away.
Indiana Jones: Put your hands down, will ya? You're embarrassing us.
Marion Ravenwood: I'm sure I wasn't the only one to go on with my life. There must have been plenty of women for you over the years.
Indiana Jones: There were a few. But they all had the same problem.
Marion Ravenwood: Yeah, what's that?
Indiana Jones: They weren't you, honey.
Question: Why did the nuke fridge scene cause so much controversy?
Answer: Another problem is that people couldn't articulate what they didn't like about the film, and point to the "Nuke the Fridge" scene as a quick example of what they think is bad about it. In the realm of Indy, it's really no more outlandish than jumping out of a plane in an inflatable raft like in Temple of Doom.
It's considerably more outlandish. The raft scene was recreated/reviewed by Mythbusters, and they found that the raft floated down at a mere 22 mph. It would be hard/impossible in real life to stay on the raft, sure, but if you buy that bit of movie cheekiness, it would definitely slow their descent enough to survive, especially since they land a) on a slope and b) on soft snow. The fridge scene, however, has no such saving grace... it's completely ridiculous and unrealistic in every detail.
Question: Perhaps I've forgotten some details from The Last Crusade in the years since I've seen it, but shouldn't Henry Sr. not be dead in this movie, because he drank from the Holy Grail which gives immortality?
Answer: Yep, you've forgotten some details. The Knight specifically states that the immortality bestowed by the Grail is limited to those who remain within the shrine. Jones Snr left the cave, ergo, no immortality.
Answer: People felt it was ridiculous and cartoonish, even by Indiana Jones standards. Even if it was possible to survive a nuclear blast via the lead lining of a fridge (it's not), or that the fridge would simply be thrown away rather than be melted/torn apart like everything else in the vicinity (it wouldn't), the impact of being flung what appears to be a mile or so through the air, then violently crashing into and rolling over the ground, would certainly kill anything inside. The controversy arose because usually, in "classic" Indy films, the fantastical elements were exactly that: fantastic, magical, and/or supernatural. This was presented as taking place in our reality, with no "power of God" or magic spells, and for many, that was just too much disbelief to suspend.