Non Sequitur Scene

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.

"This is named after the random musical number sung by a big-lipped alligator towards the end of the film All Dogs Go to Heaven. A scene that comes right the fuck out of nowhere, has little to no bearing whatsoever on the plot, is way over the top in terms of ridiculousness, even within the context of the movie; and after it happens, no one ever speaks of it again."

—The Nostalgia Chick, explaining her term "Big Lipped Alligator Moment" after using it to describe the singing goanna scene from Fern Gully. But any term you have to explain is a bad term.

Springboarding from there we find the Non Sequitur Scene (also called a "Big-Lipped Alligator Moment" by The Nostalgia Chick when reviewing Fern Gully, or "BLAM" (an appropriate term in itself, as they tend to show up with all the subtlety of a shotgun blast)) 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 Non Sequitur Scene:

Appears out of Nowhere: 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.)
Strange in Context: 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).
Never Mentioned Again: Now even important scenes are rarely mentioned later on in a film, the reason is you don't want a story to be telling you something you already knew 10 minutes ago. What this means is that upon leaving this scene the plot can suddenly start up again. Compare the Wacky Wayside Tribe.

Being merely inconsequential or strange is not enough. All three criteria have to be met. If a scene is considered "borderline" it is likely not an example.

A Non Sequitur Scene is a moment, not a subplot or entire episode (For that we have Non Sequitur Episode). Also, 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.

Compare these other tropes and consider whether an example would better fit there:

Cutaway Gag: A throwaway joke that has a Framing Device of a character reminiscing of an unrelated past story or an Imagine Spot.
Non Sequitur: A line of dialogue that doesn't follow the regular or normal conversation.
Non Sequitur Thud: A line of dialogue spoken by a character just before they go unconscious.
Gainax Ending: A similarly bizarre Mind Screw ends up being the resolution to the plot!
How Unscientific: A moment that breaks Genre Consistency, but may or may not break consistency of tone.
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.
That Reminds Me of a Song: When a character suddenly starts singing a song that has no plot relevance.
Giant Space Flea From Nowhere: A sudden Boss encounter, in video games, that has nothing to do with the plot.

Also compare What Happened to the Mouse?, Aborted Arc, Makes Just as Much Sense in Context, and Flashmob (a Real Life Non Sequitur Moment). If the Non Sequitur Scene is used to sell products, it may be a Product Promotion Parade.

Confused annoyingly often with an Arcadian Interlude.

Examples of Non Sequitur Scene include:

Non Sequitur Scenes By Medium

Advertising

You would think that it's impossible for anything as short as a TV ad to have a Non Sequitur Scene, but this ad for Herman Cain (at the time a candidate for the 2012 Republican nomination for President) proves us all wrong. It features Cain's Chief of Staff Mark Block talking about what Cain hopes to achieve by running...and then, towards the very end, out of nowhere, a shot of him randomly taking a drag off his cigarette. Combined with the closing shot of Cain slowly turning to the camera and slooooooowly smiling, the ad quickly went viral.

All The Tropes

The Real Life section above.
Any and every time an editor includes a gag or meme from a show in their entry.