When Arthur and Mera are fleeing Atlantis Arthur yells "Bogies on our 6" meaning hostiles behind them. The term for a confirmed hostile aircraft is "Bandit", "Bogey" is the term for an unknown aircraft, one that is shooting at you is definitely hostile.
When the Atlantean Soldiers come for Atlanna, when Arthur is three, she hands Arthur to his father. Arthur's hair goes from being short and curly to long and nearly straight then back to short and curly in the span of less than 10 seconds.
In the very first scene, as the storm batters the coast, lighthouse keeper Tom Curry is securing a window shutter so it stays open. If anything, he should've been making sure all the shutters were shut to protect the windows from the forces of the storm.
The propeller of the submarine is shown as generating a massive amount of bubbles, or cavitation. Submarine propellers are specifically designed to avoid this as much as possible because creates sound that can easily be detected by sonar.
When the Russian submarine is hijacked the pirate craft moors itself near the bow, presumably over or very close to the forward escape hatch. The majority of the submarine crew block themselves into the torpedo room, which is directly under the forward escape hatch. The pirates would have needed to go through the torpedo room to reach the command center.
As Arthur and Mera are sailing away from Sicily, much of their dialogue shows them at the stern of the boat, with the boat's wake and the setting sun behind them. The final shot of the sequence, from behind the boat, shows it sailing forward towards the setting sun. The sun (or the boat) has changed location by 180 degrees.
During the flight over the desert, Aquaman and Mera appear in a DHC-4 or Caribou. In the real aircraft, the cockpit is above the cargo floor level; the pilot wouldn't be able to see his passengers the way it is shown in the movie. Also, the cockpit is much smaller than depicted in the movie.
During the during the submarine fight scene, Jesse fires a 40mm grenade at Aquaman. The barrel of the launcher is rifled, imparting spin to the grenade for stability and arming. The action is in slow motion and the grenade is obviously not spinning. These grenades have a kill radius of 16 ft (5 m) and a casualty radius of 427 ft (130 m) which puts both Jesse and Manta in harm's way.
The movie shows the characters meeting in shipwreck which has the regular, "fishtank" look: greenish wood covered in barnacles, complete hulls and masts and ragged sails. While look good, wooden ships do not preserve well underwater. Any wrecks of sailing ships would be a bit more than rotten timber strewn in the sea floor, nothing remotely resembling a ship anymore.
When Orm and Nereus meet we see a stationary line of seahorses facing a stationary line of sharks. Sharks can only breath by moving to keep water flowing through their gills, so sitting still as shown would result in their suffocating. (Then again, this *is* a movie about entire tribes of humans who can breath and hold conversations in deep ocean water without an apparatus.)
If Arthur is impervious to "land made" metals, how did he get the scar above his left eye or, for that matter, get tattooed? During his youth Arthur was being trained by Vulko and was probably the one who provided the Atlantean metal to give Arthur his first tattoo. While the scar is the actor Jason Momoa's actual scar from previous incident. Arthur could've received the scar from another Atlantean metal during his training.
In the 1985 segment of the movie, when discussing names for the baby, Tom suggests "Arthur" after hearing about the current hurricane Arthur on the radio. In 1985 the "A" named hurricane was Ana, in 1986 it was Andrew. Arthur wasn't used as a storm name until 2014. But that is in our universe. In the DC Extended Universe, where history is not quite the same as ours, apparently the hurricane was named Arthur.
When Queen Atlanna has her first view of the surface world, she acts like the cliche "ignorant savage," stabbing a television as if it's a foul demon. But we soon learn that she's from a world which is far advanced beyond anything we've ever known. A 1980s-level television should be as primitive and unremarkable to her as Thomas A. Edison's original phonograph is to us.
The goof items below may give away important plot points.
The Sahara Desert has been a desert for 2-7 million years. Rome was founded in 753 BC. There is no way a message from "before the Sahara was a desert" could refer to a statue of Romulus erected in Sicily, which itself was not under Roman control until 241 BC.
Near the end, when Arthur fights Orm for throne and pushes him against the propeller, Orm's cape gets ripped off you can also hear the sound effect, but when Arthur throws him it's back on again, and finally the cape is gone when Orm gets up.
When Mera leaves the gladiator arena, she quick-changes from party dress to active wear. Yet when she rushes to Arthur's rescue she is still wearing her bracelet, which she would have had to remove in order to take off the party dress. It makes no sense for her to re-don the bracelet when time was a-wasting.