In the final episode, "Turnabout Intruder", insane Janice Lester switches bodies with Captain Kirk so she can stop being a woman and become a Starfleet Commander. You'd Expect: She would be very careful not to do anything that would give her away, although she seems fairly confident that she can pull it off. Instead: She makes official log entries as the captain, where she actually brags about how she's duping everyone. There is absolutely no good reason for her to do this. Only the fact that she's essentially insane can excuse her.
From "The Alternative Factor", two of the Enterprise's dilithium crystals have been stolen, and Kirk immediately suspects Lazarus, who, earlier that day, begged Kirk to give them to him so he could kill his enemy. Also, Kirk and Spock have been growing increasingly suspicious of Lazarus' true intentions. You'd Expect: The crew to keep constant watch on Lazarus, who, by this point, is resting in Sick Bay. McCoy even says Lazarus is "not going anywhere. Not this time." Instead: McCoy and the others leave Sick Bay, leaving no one to keep guard on their potentially dangerous guest. Sure enough, Lazarus leaves Sick Bay and takes the other two crystals.
Later on in that episode, Kirk is accidentally transported to an antimatter universe, where he meets a parallel version of Lazarus who, unlike his prime universe counterpart, is completely sane. Anti-Lazarus confirms what Kirk and Spock have feared; that if he and Lazarus come into contact, it would set off a chain reaction that would destroy both universes. You'd Expect: Assuming we take the rather spurious science at face value, that Kirk would stun Lazarus, drag him back to the Enterprise and then have the Enterprise destroy Lazarus's spaceship, which would sever the link to the antimatter universe. Alternatively, just kill Lazarus, which would be a justifiable action given that two whole universes are at stake. Instead: Kirk and Anti-Lazarus come up with a plan whereby Kirk throws Lazarus into the dimensional link, where Anti-Lazarus will hold him. The Enterprise then destroys Lazarus's spaceship, ending the threat from him... but also consigning Anti-Lazarus to spend the rest of his life (if not the rest of time) having a fistfight with his insane counterpart.
In "Datalore", the Enterprise happens to find Lore, an identical twin robot of Data. At one point, when Data and Lore are alone, Lore reveals himself to be an Evil Twin by incapacitating Data, then claims that he is Data and that Lore attacked him, and he disabled Lore in response. You'd think: That the very blatantly obvious fact that the two are identical would make Picard suspicious, and he would ask Lore something only Data would know to find out if he was really Data or not. Even if he didn't bother with any of that, you'd think he'd at least be sure to keep a careful eye on Lore and take any advice from him with a grain of salt, just in case. Instead: Picard implicitly trusts Lore, believing he's Data, even when he does things that Data wouldn't do. Even worse, Wesley explicitly tries to point out the possibility to Picard, and Picard for some reason ignores him. Sure, he's a Creator's Pet, but that does mean he has a tendency to be right. The only reason everyone on the Enterprise didn't die due to Picard's appalling stupidity is that Wesley goes against orders and manages to save the day.
In "The Price," The Federation and several others are engaged in a bidding war over what appears to be the only known stable wormhole in the galaxy. Picard, however, is not fully convinced, and sends a science team through the wormhole to verify its stability so that they won't end up with a proverbial lemon. You'd Expect: The negotiations to be put on hold until the wormhole is declared safe. Instead: Not only do they continue, but one less-than-scrupulous representative manipulates everything in favor of his clients; he ends up winning, but the science team then returns to reveal that the wormhole is in fact dangerously unstable. The representative's clients are not pleased.
Another moment occurs during this debacle: The Federation's science team (consisting of Geordi and Data) is accompanied by a Ferengi team. Once they both go through the wormhole, Geordi and Data discover that they're not where they're supposed to be. It turns out that the endpoint of the wormhole shifts location every so often, and Geordi's sensors indicate that another shift is imminent. He immediately warns the Ferengi that they need to get the hell out of there now. You'd Expect: The Ferengi would listen. One of them is a scientist, after all; he should be able to verify what Geordi's telling him. Instead: The Ferengi tell Geordi to shut up. As a result, Geordi and Data make it back into the wormhole just before the endpoint shifts, trapping the Ferengi in the Delta Quadrant.
In "The Perfect Mate," the Enterprise is transporting an ambassador to a peace conference. Said ambassador has brought along a gift for the other side's chancellor as a peace offering, which he describes as "quite fragile and quite irreplaceable." You'd Expect: Picard to seal off the cargo bay and post security guards. Especially with two Ferengi aboard. Instead: He doesn't. One of the Ferengi gets into the cargo bay and screws everything up.
In the two-parter "In Purgatory's Shadow"/"By Inferno's Light", Garak and Worf are captured in a runabout and taken to an asteroid prison. You'd Expect: Their ship be impounded, disassembled, or outright destroyed. Instead: The ship is left, unguarded and completely active, in transporter range of the asteroid with no other ships in the vicinity. Escape is as simple as calling the runabout and having it beam them to safety. For Added Stupidity: This was actually brought up in a later episode when Sloane is auditioning Bashir for Section 31, which means, either then or in hindsight, even the writers knew it was contrived.
An earlier episode of The Next Generation had a wormhole with a stable entry point on one side, and a constantly jumping exit point on the other. Voyager finds this wormhole during the second season. They also find the Ferengi that were stranded in the Delta Quadrant, who have set themselves up as gods among a bronze-age people. You'd Expect: That they would make a beeline for the wormhole, or at least grab the Ferengi first and then hightail it back to the Alpha Quadrant, letting 5 or 6 generations of the bronze-age people undo all the damage that was done. Instead: They try to fix all the meddling of the Ferengi (itself a Prime Directive violation, arguably), who escape and even destroy the wormhole.
Future Janeway plans to negate her own timeline by helping Voyager get back much earlier than intended. She'll have to break a lot of rules to accomplish her plan, though. You'd Expect: That any of the people who indirectly/directly help her mission (Barclay, Miral Paris, Harry Kim) would have gotten some sense and realized that Janeway would erase the past twenty-plus years of their lives if they allowed her to continue through with her plan. Instead: Barclay and Miral unquestionably go along with the plan (Miral even tests the device to make sure it works!), and even Harry Kim is somehow swayed after Janeway talks to him. Basically, everyone in the future has to act like an idiot in order for Janeway's plan to work.
Weighing Janeway's need to get back home ASAP with the lives Voyager could have potentially saved over the next several years makes her choices seem more suspect. Janeway is told point blank by her future counterpart that, over the next two decades, she'll only lose 26 crew members (which is an average of about 1 per year), but that Voyager will have met and helped countless races all the way to the Alpha Quadrant. Either way, though, Tuvok will still end up going insane - no one can do anything about that. You'd Expect: That, knowing this information, Janeway could have come back home as a legend and still kept Chakotay and Seven alive by not assigning them to away missions. Plus, the ship now has advanced Borg armor that would repel most enemy attacks. Instead: She (with the help of her counterpart) destroys a Transwarp Hub and sails right on home, content in the knowledge that she's saved a few more crewmembers at the cost of thousands - if not tens of thousands - of people who would potentially benefit from Voyager's assistance. Not to mention all the technology and information they would pick up along the way.
In the episode "Timeless", Harry Kim and Tom Paris manage to build a slipstream drive like the one on the alien ship from "Hope and Fear". Problem is, it destabilizes after a few minutes, so they have to make constant course corrections. Harry tries but can't keep up, killing the entire crew except himself and Chakotay. A future version of Harry Kim rewrites the past so that the ship drops out of transwarp after two or three minutes in its trial run, so that it doesn't crash and kill the crew. You'd Expect: Harry to realize the technology works in short intervals, and use it to "puddle-jump" the ship all the way to Earth. After all, if you have a proven window of stability, then you can just stop before passing that window. The crew (including Harry) know this fact for certain, and discuss it at length. Instead: Janeway decides the technology is too dangerous and orders it dismantled, while being disappointed that their experiment didn't work. But it did work! You just cut ten years off your journey, and were seconds away from making it back home!!!
"Innocence": The crew finds a planet were the people claim the "children" they have are aging backwards and so they want them back. You'd Expect them to be suspicious after the fact the aliens tried to kill them at every turn, refused to cooperate and provide no proof. Instead they hand them over. For Added Stupidity This already happened to the Feds... and it turned out to be an intelligence test (TAS novel).
In "Someone to Watch Over Me", the Doctor and Paris have a bet going to see if he can teach Seven how to go on a date without being her usual overbearing self. After an early attempt is messed up by Seven tearing the guy's ligament during a dance, the Doctor takes her himself for the "final exam", so to speak, which is a dinner being held for an alien ambassador. The Doctor's teachings work, and Seven does splendidly. You'd Expect Paris to wait until after the dinner to settle their bet. Instead He does so when Seven is literally standing right next to him, forcing the Doc to admit to betting on her performance. She is righteously pissed, and storms out. For Added Stupidity This can't even be excused as Paris being vindictive about losing. He seems to have completely forgotten Seven was standing there, as he hastily tries to take responsibility when she gets mad about it.
The Xindi in their arc of season 3. They hate humans, they are building an Earth-Shattering Kaboom gun. Now at this point they have five major advantages: Their enemy has no clue they exist, they have four hundred years to refine their prototype, they have allies who give them technology and can see the future, they live in a remote and inaccessible part of space, and they can travel nigh-instantaneously across the universe. Now they complete a Small Country Shattering Kaboom prototype of their weapon, and... You'd Expect: They test it on some out of the way moon or planet no one will miss. Then they use the data from that test to refine their final version, teleport it over to Earth, and destroy the planet with one shot. Instead: They test the prototype on Earth itself. Earth immediately sends The Enterprise after them, which: finds them, destroys their next prototype, convinces them not to blow up Earth, and murders their future-seeing allies. Good job, Xindi! You failed only because of your own stupidity. For Added Stupidity: They were testing prototypes at remote and secret locations! Using the prototype on Earth was not only stupid, it was also completely redundant.
In "These Are The Voyages", the crew has just rescued Shran's daughter from a group of his former business associates. As the crew are heading back home after the rescue, Archer and Tucker hear a message over the ship's intercom (delivered by T'Pol) stating that a group of intruders - the same people who kidnapped Shran's daughter - have beamed onboard. You'd Expect: That Archer and Tucker would wait for security to deal with the intruders, or tell the bridge to teleport the intruders out/attack the ship to prevent them from leaving. Instead: Archer and Tucker rush out to confront the intruders, sans weapons. They predictably get captured immediately by the aliens. Tucker (who has been shown to be capable of talking down/stalling villains in the past) tells the aliens to knock out Archer (while security is nowhere to be found), and then leads them to a room where he blows himself and his captors up, presumably dying afterwards due to injuries sustained. There was a Fix Fic written later on to Retcon the silliness of this incident as a ploy by Section 31.
"A Night In Sickbay": Trained diplomat Archer is negotiating with a race who are so prudish they have in past stormed off the ship after discovering humans eat in public, to access a component that is crucial to the continued functioning of the warp drive. As part of this, he goes to visit a stand of sacred trees. You'd Expect: He would go out of his way to be as polite, dignified and sensible as possible. Instead: He brings along Porthos, who, no matter how trusted a companion he may be, is still a beagle. Even a small child could have predicted the outcome of this action.
Terrell and Chekov arrive at Ceti Alpha V, on the Reliant, thinking it was Ceti Alpha VI. They go down to the surface. Chekov finds out that the wreckage on the planet was the Botany Bay. Unfortunately, Chekov and Terrell get captured by Khan and his minions. Back on the Reliant, two of the crew members are trying to contact Terrell, but are getting no response. You'd Expect: The crew two crew members to go: "Something's wrong. Send someone down, and find Terrell and Chekov.", and potentially find out that Khan was on the planet and has Chekov and Terrell hostage, and save them and potentially stop Khan. Instead: They merely shrug and say, "Let's give them a little more time.", which allows Khan to brainwash Chekov and Terrell, and take control of the Reliant without warning.
The Enterprise first encounters Reliant after sketchy reports that should suggest that something is seriously wrong with Reliant's handling of the Genesis situation. After hailing her numerous times with no response, Enterprise receives an excuse that a critical communications component is faulty, an assertion that does not survive a cursory scan of the ship by Spock. So now, whoever's on that ship is both acting suspicious and out-and-out lying. You'd Expect: Kirk, the combat veteran, who's probably trained extensively for these kinds of situations, heeds the sage advice of Saavik, the fresh-out-of-Academy cadet, and raises the shields until the situation can be clarified. Instead: He, and the allegedly intelligent Mr. Spock, shut Saavik up and blithely keep going on towards Reliant with shields down and weapons disarmed. Khan and his crew knock the stuffing out of Enterprise.
In Star Trek III Spock's father, Sarek, tells Kirk that Spock's body should have been returned to Vulcan, not left on the Genesis Planet; if they don't retrieve the body, Spock's katra will be stuck in McCoy's head, effectively killing them both. You'd Expect: That Starfleet, when informed that the Vulcan ambassador is understandably furious that his son's body wasn't returned home according to the rules of their culture, and that an officer's life or sanity is at stake, would fall over themselves to get in touch with the ship that's already in orbit around the Genesis Planet and ask them to take five seconds to beam Spock's coffin aboard. Instead the admiral flatly refuses to do anything, throwing in a patronising comment about how he doesn't understand 'Vulcan mysticism', and is later amazed when Kirk and co steal the Enterprise and make for the Genesis Planet anyway.
At the start of the film the Big Bad, Shinzon has two immediate objectives: the first is to abduct Captain Picard and drain him of all his blood to cure Shinzon's rapid ageing disease — a disease which will kill him in a week at the most — while the second is to have Data's dimwitted prototype, B-4, steal the Federation's defense data from the computer banks. B-4 is picked up on a planet near the Neutral Zone, and the Enterprise proceeds to the Romulan homeworld of Romulus. You'd Expect: Shinzon to have B-4 steal the relevant data during the flight over to Romulus, or as soon as he reasonably can once they've arrived. Then Shinzon can beam Picard and B-4 aboard his flagship, the Scimitar — which can fire while cloaked and has defensive and offensive capabilities around three to four times that of the Enterprise — and blow the hell out of the Enterprise before the rest of her crew can work out what's going on. Instead: He has the Enterprise sit around for half a day, seemingly without accomplishing anything, then waits another whole day before having Picard join him for dinner, and then finally abducting Picard another half-day or so after that. During the intervening time, the Enterprise crew are able to detect the planet-killing thaleron weapon aboard the Scimitar, and realize that something's wrong with B-4 (who they swap for Data when Shinzon actually tries to retrieve him).
During the initial meeting with the Enterprise crew, Shinzon is very obviously taken with Counsellor Troi, to Stalker with a Crush levels. You'd Expect: Shinzon to remember that there are more important things at hand than his dick, and get the hell on with his plans to save his life and destroy Earth. Alternatively, if he's really that desperate to have Troi, then just abduct her along with Picard and B-4. Instead: He uses his viceroy's psychic powers to Mind Rape Troi while she's making love to Commander Riker, thereby confirming to the Enterprise crew (as if they didn't already have enough reason to suspect it) that he's evil.
The Enterprise has been boarded by light-sensitive Remans during a Red Alert, when the ship's lights are dimmed. The Remans are not wearing goggles. You'd Expect: The bridge crew laugh and beam the boarding team into the brig, since they still control the transporters. Instead: Long, drawn-out running phaser battle through the corridors of the ship ensues. You'd Expect: Someone to turn the lights up, blinding the Remans and ending that threat. Instead: Long, drawn-out running phaser battle through the corridors of the ship ensues. You'd Expect: The bridge crew let the security teams do their job, and keep their focus on the space battle going on outside. Instead: Important crew members, including the first officer and the guy in charge of the ship's weapons, abandon their posts on the bridge in the middle of a fight to take part in a long, drawn-out running phaser battle through the corridors of the ship.
The baddies want Picard. The good guys beam him over, and the transporters promptly fail. They do, however, have an prototype emergency transporter. You'd expect: The good guys to beam over a bomb, use the independent transporters in the shuttles, have Data/a security team with a tech on it take a shuttle and hack their way in, or replicate the emergency transporter. Instead: Data jumps for the enemy ship, finds Picard, slaps the transporter on him, then dies in the most pointless Heroic Sacrifice ever.
So things with Shinzon have gone irreparably pear-shaped, forcing the Enterprise to fall back to Federation space. They call ahead to have a fleet of ships rendezvous with them to engage Shinzon's Scimitar. You'd think: the Enterprise crew would plot their course carefully, and avoid anything that could interfere with their ability to contact the fleet for help should they get ambushed by Romulans—or worse, Shinzon— on the way. Instead: They sail headlong into a dense nebula, which scrambles astrometric scans and long-range communica-..."WORF! SHIELDS UP!!"
Star Trek Into Darkness: Admiral Marcus decides the frozen bodies of seventy-two genetically engineered superhumans is a liability and plans to get rid of them. Said Admiral has a massive secret base on one of Jupiter's moons and has moved huge amounts of resources to and from it in secret, and he's apparently in charge of weapons development. It also turns out that the leader of the superhumans, Khan hid the bodies in special long-range photon torpedoes. You'd Expect: Marcus just take them back out. If Kirk's crew could do it without the specifications, surely his men could having that information. The bodies could be disposed of any number of ways that wouldn't arouse suspicion, for instance launching them into Jupiter from his secret base there. Instead: He gives the torpedoes to the Enterprise and orders Kirk fire them at the Klingons. Since he can't have people finding the bodies inside he has his subordinates vigorously refuse all attempts at opening one up, antagonizing Scotty and virtually guaranteeing somebody will decide to open one on general principles. This eventually leads to Kirk discovering the truth, and gives enough credence to the captured Khan's story that Kirk turns on Marcus.