
Stupidity: Natasha is in Norway on the run - Norwegian radio mentions her by name, Yelena talks about her being on magazines, she's very recognisable, and yet she goes into a gas station with her sister without changing her appearance in any way, and loudly talks about being an Avenger and killing people, right in front of the gas station attendant who for all she knows would be on the phone to the authorities the second she's out the door.

Stupidity: The coast cities should have an alarm that detects the Kaijuns. If they do, they should work when the beasts are few miles away from the shore in order to give the people time to get to the shelters. Instead of that there are still people on the streets in Japan and on the docks in Los Angeles (in the flashback) while the kaijuns arrive from the sea.

Stupidity: When the crew think that Dorothy is a robot with a bomb planted in her and is capable of exploding, they begin to think that she is about to explode and attempt to dodge it. Rather than hide somewhere or take cover behind an area that may protect them, they just get down on the floor in front of them and cover their heads with their hands. If they thought the explosion was going to be as deadly as the one they saw on the news, this method of taking cover wouldn't be even remotely effective. (00:24:40)

Stupidity: When the Pteradons are loose and flying over the crowds, people run from inside buildings out into the open where the birds are attacking people. (01:22:00)
Suggested correction: A person is smart, people are stupid. A crowd in a panic would do exactly that sort of thing. Like trampling over each other running in panic from a shooting or a fire when not needed. It's the nature of panic in humans as a group to act stupid. It's not a movie mistake. It's actually one of the more realistic parts of the film.
Suggested correction: The dinosaurs were also breaking into/attacking people inside, so they weren't safe no matter what.
True, but one option is certainly safer than the other.

Stupidity: All the mutants on the island need Magneto at the last second to take control of the missiles so they don't land and explode. Why didn't Azazel just teleport everyone off the island?
Suggested correction: Given the situation, there was probably very little time to come up with a plan like this. The missiles were flying towards them extremely quickly, and there would have been nowhere near enough time to band every single person together for Azazel to teleport them all out of there in just a few seconds.
There was plenty of time. He could have taken 5 at a time down the Beach 3 miles and came back for the rest. Earlier in the movie he was moving so fast men were falling from the sky at nearly the same time and he was teleporting them one at a time.

Stupidity: After hearing the question on the radio station to win the Turbo Man doll and a failed attempt at the pay phone to answer it, Howard and Myron end up running two blocks to the station to answer the question. This is really stupid because the prize was going to the first caller with the answer and in the few min it would take to run to the station two blocks away and finally reach the DJ, many calls would have come in and most likely someone would have won already, way before the men arrived there.
Suggested correction: This isn't a stupidity or a mistake. What else did you expect them to do? Plus, the DJ even later asks if they expected him to actually have a Turbo Doll in the studio, and they both said "yes."
Not only that but, Myron had ripped the phone receiver out so Howard couldn't give the answer to the DJ. Howard and Myron running to the radio station was the only other option they had. At that point, it would only be a matter of if they got to the station in time before somebody could call in with the right answer.
It was a very easy question naming santa's reindeer. Although some people may not quite know them all... most people do. Now if it was a relatively hard question, our two guys probably had a better chance. But all good points made here in all these comments.

Stupidity: The prison where Magneto is held should have had a contingency plan just in case of escape, like machine guns outside the cell, or tranquilizer darts, or gas.
Suggested correction: What good would machine guns and tranquilizer darts be against a man who can literally control and manipulate metal? As for gas, the area outside the cell is so large that any gas would quickly dissipate before it could be effective.

Stupidity: Any fighter pilot should know better than to attempt a low flying attack so close to a hill, let alone into a tunnel. How could he not know that it was there? How'd this moron even get his pilot's license?

Stupidity: It was announced on the radio "to stay inside your homes and to only leave in a state of absolute emergency." Then the house shook from an earthquake. May's mom (Gran) called and said, "I'm not sure you should come right now... the roads aren't safe" and "it's not safe to be driving around out there, especially with the kids." May, who told her mom she had the trip planned, finished packing and said, "Kids, we're leaving." Before driving very far, a "monster" was near her car. (00:01:33 - 00:03:47)

Stupidity: When Jonathan is chasing the truck to try and save Sarah, he could have removed his heavy backpack and run faster...but instead just leaves it on. No wonder he wasn't fast enough.

Stupidity: The highly trained commandos who knew everything about the NSA base and penetrated it during a perfectly planned operation through weapons and devices that pierce steel like butter, decided to stand outside the barn (so they knew something valuable was there) and its wooden door. It even has a direct line to the command center.

Stupidity: Billy and Allie saw there were several men in the area also searching for the medallion and the kids even hid from the men minutes before finding the medallion. Instead of quickly sneaking away after they found the medallion, Billy and Allie stayed in the same spot and Billy shouted, "We found it! We found it!" Seconds later, two men came from behind the tree. (00:24:00 - 00:24:21)

Stupidity: When Governor Swann is trying to encourage his daughter while she's locked within the ship's chambers trying to escape, she climbs down to a boat. Nobody in the Navy would have left a boat there, neither would Jack (and why would he, especially without anyone noticing?). The boat just mysteriously appears to help her escape.

Stupidity: When the flying snowmobiles attack, Bond tells Elektra to go into the gully, and he'll lead them into the trees...completely ignoring the fact that these bad guys are here to kill her, so why would they follow him instead? Of course they all do. Either a plot contrivance or a blatant set up, which he doesn't remotely consider.

Stupidity: The heroines guess the plan of the villain with a thought process that can only be explained by the fact that they read the movie script. There is no way for them to surmise that the attack on the Skrull base, perpetrated under subterfuge, was intended to "syphon away the atmosphere", and not just the usage of a really powerful weapon. Even assuming that Kamala's objection to this particular point ("they could have just invaded or used a bomb") is a logical explanation that would disqualify without a second thought the fact that a Kree warlord would use a terrifying and cost-free weapon that is also a symbol of Kree's culture just for its power, and so that it makes sense that the heroines just instantly commit to an alternative theory, there's still a leap of logic in assuming the exact next step for the Kree's quest on revitalizing their homeland (which incidentally involves randomly dumping out of the blue a gigantic quantity of salt water on a densely populated planet).