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A Non Sequitur is a bit or line of dialogue that is intentionally out of place, usually designed to elicit a comedic reaction. They have no actual bearing on the plot, although they are staples of characters who are part of their own little world like the Cloudcuckoolander or The Ditz.
Springboarding from there, we find the Big-Lipped Alligator Moment, or BLAM (an appropriate term in itself, as they tend to show up with all the subtlety of a shotgun blast). This is a very bizarre scene in an otherwise normal story that veers off into the surreal or strange. Upon exiting that scene, the plot continues on like it never happened.
There are three precise criteria for measuring a Big Lipped Alligator Moment:
The plot comes to a halt as the scene takes its spot in the running time. There can't be any Foreshadowing and it can't take a logical place in the plot (e.g., coming across a trap while Storming the Castle is not random, but expected.)
The fictional setting, characters and narrative devices have to be at odds with the scene. In that regard, World Building moments, strange personalities, and a surreal story structure that can explain its origin are exempt from this trope. For instance, All Just a Dream gives a good reason, as using the trope tends to go hand-in-hand with the surreal (although for some genres and franchises, using the scenario of All Just a Dream itself may qualify).
Simply being random, strange, or inconsequential to the story is not enough. All three criteria have to be met. If a scene is considered "borderline", it is likely not an example. BLAMs are often Level Breakers as well. Usually, this is done just because.
The Trope Namer is All Dogs Go to Heaven, and the Trope Codifier is The Nostalgia Chick and The Nostalgia Critic from their review of FernGully while commenting on another example. In the trope naming scene (in a movie that mostly deals with talking dogs, the afterlife, and mafia undertones, somewhat strange itself) a big-lipped, Cajun-accented, bone-through-the-nose alligator takes a liking to the main character Charlie and forces him to sing a duet "Let's Make Music Together". While the alligator goes on to have a role in the plot later on, Charlie is noticeably very confused over the whole song.
The Nostalgia Critic and The Nostalgia Chick have had to post supplementary videos on this topic, due to confusion on what does and does not count. One is that it is a moment, not a subplot or entire episode (For that we have Bizarro Episode). And they emphasized that it has to go against what is considered normal.
Context and the nature of the situation matters immensely in comparison to the sliding scale of realism used by the story in question. This trope can be objectively observed but because of the dissonance between the audience and the characters (the audience doesn't live in the exact same world, so what is strange to the audience might be commonplace for the characters) this often ends up as a debatable topic. Even the trope namer has been subject to debate.
Adding to the controversy is the fact that, in certain circumstances at least, belief in this trope could be considered an example of the Perfect Solution Fallacy; defined in this case as excessive pedantry about adherence to the Law of Conservation of Detail. Aside from anything else, there are works where irrelevant or obfuscating details are actually important; pare down the details too much in a murder mystery, for instance, and you risk making the plot obvious from the outset. Being excessively paranoid about avoiding this trope or conserving detail, also potentially puts constraints on creativity in general terms, as well.
Compare these other tropes and consider whether an example would better fit there:
Brick Joke: It comes out of nowhere, seems like it has no relevance at first (or only marginal relevance), disappears for awhile then comes back like a boomerang later on at some random point and becomes relevant, even if only to a very minor plot element. If it does so more than once, it becomes a Running Gag.
Disney Acid Sequence: A surreal visual and musical scene that may or may not have plot relevance; because of the occasional lack of relevance there tends to be overlap. Is usually just a chance for animators to Show Their Work.
During Nintendo's November 13, 2013 Nintendo Direct, the first shot of Reggie Fils-Aime's office is of his giant stuffed Mii head resting in a chair. The camera then pans over to the real Reggie off to the side as he begins his preview of upcoming games. The Mii head has no connection to the games being shown and is never seen again in the Nintendo Direct.
GEICO commercials seem to be using this as a running gag as of "Did you know that...?" , hereare but two examples.
In the Norse Mythology tale of "The Death of Baldur", there is a part where the Ęsir are gathered at the funeral pyre of the recently-killed Baldur, when a dwarf shows up. The dwarf, named Litr, casually strolls in when Thor notices him and punts him into the funeral pyre. The tale then resumes as normal and the dwarf is never mentioned again.
These happen all the time in real life due to its general lack of a coherent plot.
This is true for almost any signal intrusion or broadcast piracy incident. While some attempt to convey some sort of message, a good number are intrusions for the sake of intruding, such as the above-mentioned incident or simply designed to scare people.
The second one also happened while an episode of Doctor Who was being run. Just picture this: You're watching the show, and then suddenly, a guy wearing a Max Headroom mask just appears on the screen and does all these bizarre and offensive things. Then back to Doctor Who as if nothing happened. And you're left sitting in front of the TV thinking "What the Hell?"
Dreams. And 99% of the time, not only do you not mention them ever again, but you forget that they even happen.
Musical numbers and certain performances in film and live-action TV up until the late sixties. The plot would stop for a performance, then when the performance was over, the plot would go on as though nothing had ever happened. This was done to make it easy to cut a number for certain markets that might find it offensive. For example, black performers might be cut for certain Southern states.
At a televised golf tournament, Tiger Woods went to tee off, surrounded by hundreds of spectators who were quiet per standard gold procedure. As soon as the thwack of the club was heard and Woods' ball was airborne, someone in the gallery shouted "MASHED POTATOES!" - the TV announcers were bewildered by it.
In the 2013 National League Championship Series, before the start of Game 6, Cardinals pitcher Joe Kelly and Dodgers utility player Scott Van Slyke got into a 12-minute staring contest. Both men remained standing on the playing field outside their dugouts, not having moved from their positions where they stood at attention for the national anthem. This went on while the grounds crew was busy getting the field ready, and Joe Kelly's teammates were going through final warm ups on the field. The home plate umpire finally had to gesture for them to get back into their dugouts.
Dino Attack RPG has the scene where Trigger dies and ends up in a strange hotel run by a strange character known simply as "the owner" who may or may not be a pervert with blue hair and no pants. He then leads Trigger through a variety of places in the hotel, all occupied by characters who have died in the RPG and construction workers allegedly responsible for the Big Bang, before he realizes that for he temporarily has to share a room with his arch-rival Silencia Venemosa. Though part of the initial build-up had been used before (and was intended to be satirized here), this sequence was never mentioned again and is so strange even by the standards of the RPG that its status in canon isn't even 100% clear.
There was also the part where some people became Toa from Bionicle, and others started using exo-suits. It became known as the "Bionicle meets Exo-Force RPG" and was never spoken of again, except in hushed whispers in dark street corners.
The rather bizarre argument that used to be in the TV Tropes section of the The Fourth Wall Will Not Protect You article. Note that this type of edit counts as Natter and nowadays could get you an angry talking to from the mods or a suspension.
Oh, and that image at the top of the page isn't just a picture, Pinkie Pie actually is crawling out of it. I'd run now if I were you.