Please make sure the moment is fictional and is neither an event that occurred in real life nor something gameplay-related. We have a perfectly good Scrappy Mechanic page for the latter.
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theenglishman: There's one absolutely groan-worthy moment in L.A. Noire during a flashback to Cole Phelps' memories of Okinawa. Cole, then a Marine officer, is trying to control his squad's bloodthirsty desire for Japanese casualties, and asks them why they're even fighting the Japanese in the first place. One says it's because the Japanese attacked Pearl Harbour, but Cole responds that the Japanese attacked Pearl because the US had cut off their oil supply to Japan. Then, with almost a wink at the audience, he wonders what they would all think if the US ever invaded another country for their own lack of oil. Realistic or not, it was such an awfully blatant Take That that it ruined the entire scene. That being said, the rest of the game is so amazing that the amount of damage caused by this ultimate moment of suck is minimal at worst.
egregious: Phelps' affair with Elsa. There is absolutely nothing to establish Phelps' relationship with his own family beyond a passing comment that he even has a family at all. His initial interaction with Elsa is through questioning about the stolen Army surplus goods and she's rather curt throughout the whole line of questioning. Literally the next thing that happens is the player tailing her cab to her apartment, she and Phelps sleep together, and he is demoted to Arson at the end of The Manifest Destiny case. Considering the game builds Phelps to be a By the Book Cop, this is quite a 180 degree turn for him..
Regu: The end of the homicide storyline ended on a very sour note that left a bad taste in my mouth. To elaborate, you spend an entire case figuring out who the Werewolf killer is and where he's hiding. The case itself is thrilling and makes you think by reading excerpts from famous poems to discover the next clue. After going through a maze, walking over the tar pits, climbing a to the very top of the library tower, and most dangerously, tight rope walking to a chandelier that is very high up to find a clue, and then almost falling with it when it collapses, you find the killer at an abandoned church. after a rather lackluster chase, you kill the Werewolf, only to have the Captain Donnoly tell you that his identity can't reveled to the public and the case will be closed... simply because the guy is the half-brother of a influential politician. The only good part is that the people whom you've wrongly accused are set free.
Illuminatus: Manifest Destiny. The LA media learns at the beginning that the LAPD has been taking payoffs from a madam and basically running their own prostitution ring. Desperate to get out from under this, they jump at the opportunity to smear Cole when Roy Earle reveals Cole's affair. The problem is, the case itself is the climax of the side plot wherein Cole's former Marine underlings get so furious with his success that they rob the Army of crates of guns and drugs and become gangsters of their own - butting heads with the real-life Mickey Cohen. The case itself is a citywide bloodbath as Cohen tries to assassinate the Marines all at once, including attacking a public bus with a machine gun and murdering two of them right on the red carpet at the Chinese Theater. You also learn that the mafia has a mole inside the LAPD who fed them these names, leading to the deaths of most of Cole's old unit and indirectly to the death of several cops. Finally, the case reaches its high point when Cole arrests and confronts Courtney Sheldon, the mastermind of the heist and the man who shot him in the back on Okinawa, only for Capt. Donnelly to interrupt this pivotal scene to berate you for the affair and demote you to Arson. Almost none of these massive plot threads are mentioned again.