There are several in The Room. The most infamous being this (which, depending on how much credit you give to the screenwriter, might qualify as a deliberate example of Hypocritical Humour):
Mark: "How was work today?" Johnny: "Oh, pretty good. We got a new client at the bank. We'll make a lot of money." Mark: "What client?" Johnny: "I can not tell you, it's confidential." Mark: "Oh, come on." Johnny: "I can't. Anyway, how is your sex life?"
Played as a joke in Time Bandits when Robin Hood asks a dwarf bandit how long he's been a robber. The robber replies, "Four-foot-one!" A confused Robin stammers out, "Four-foot-one? That... that... that is a long time, isn't it?" before moving on. Since the bandit is an ageless, time-jumping angelic being, he apparently decided to simply supply his height rather then try to answer the question.
Groucho: "Well, art is art, isn't it? Still, on the other hand, water is water. And east is east and west is west, and if you take cranberries and stew them like applesauce, they taste much more like prunes than rhubarb does."
Live Action TV
Similar to the Scrubs examples below, the Sherlock episode The Sign of Three has Sherlock delivering a speech at John's wedding. While he's talking, he realises that someone in the room is about to murdered, and goes on a deductive rampage in his mind trying to figure out who it is and who's going to murder them while also trying to keep his speech going and not arouse too much suspicion from his audience. The result is a semi-coherent speech to a crowd of confused people that keeps dipping into non sequiturs before Sherlock manages to shakily work whatever he just blurted outfrom his train of thought into the speech.
Pablo (Spanish accent): One time I threw a stick at a monkey...
Subverted and Lampshaded in Scrubs. J.D. always appears to the other characters to be saying these, because he constantly imagines something related to the issue at hand and then makes a comment about his fantasy that winds up sounding totally nonsensical to the other characters. Generally, the longer and more elaborate the fantasy, the more removed his comment will be from the situation that triggered the fantasy in the first place.
Precious Roy: This is Precious Roy, and you kids better pay for that lap dance!
Sifl: Precious, we're talking about the Civil War Corpses...
Precious Roy: APPLE... CIDER!
Sifl: ... What?
Precious Roy: Suckers!
In the song by Andrew Huang, each word is a non sequitur, even though there never had been a real point to rabbit-trail away from other than the statistics at the beginning. This how many of his other songs tend to be.
The music video for Rich Boy's Throw some D's is about money, women and cars with modified classic cars being the primary focus. After about half way they start to inexplicably cut away to shots of a guy sitting on his front porch in a wheelchair and even more confusing is at the 3:19 mark when it cuts away to a guy just sitting in a shop. Most other cutaways are just displaying the ghetto and what different kinds of people live there, and it's a common thing to see in most music videos but these ones do feel a bit out of nowhere and thus funny.
Throughout the series, NPCs can have conversations with each other. However, these conversations consist entirely of random voice clips, so they are literally nothing but a series of strung-together non-sequiturs.
Scarface: The World is Yours is stuffed with lines of dialogue. That sometimes doesn't flow together quite well. Tony would ask a girl to clean his pool and she demands they go shopping. Then he asks why she hangs around all the time, even though he invited her to live there. Then Tony starts talking about murdering his enemies, sometimes even after they are dead.
raocow: You are not the unicorn of my love, football Charlie! You are merely a pawn in the great game that we like to call... Mario World. Although you are one of the important pawns. You are like, that one pawn in the middle that you start in the beginning of the game in order to, like, capture the whole world and stuff.
Nappa from Team Four Star's Dragon Ball Z Abridged says one of these with nearly every sentence. Much to the confusion and frustration of everyone around him. It is even lampshaded in the Dragon Ball Z Kai Abridged video.
Nappa: Hilariously derailing one-liner.
Goku, though not as frequent, at times does this as well.
About 50 percent of everything Ed says from Ed, Edd n Eddy says is a non sequitur. Often, when he's asked a question, his response is "buttered toast" or "gravy" and he randomly shouts "I love chickens" even when there aren't any chickens present.
Gerald: (singing, after everyone else has stopped)Courteney Cox, I love you; you're so hot on that show...
Kyle: Dad... Dad? We're singing about a dreidel.
SpongeBob SquarePants has some as well. One of them involves SpongeBob trying to get Mermaidman and Barnacle Boy out of the retirement home. He attempts to reason with them, then Mermaidman gets fed up with it and says the following:
Mermaidman: "If you don't get out of here, then by the power invested in me, I now pronounce you man and wife!" Orderly: (walks into the retirement home's main room) "What is going on in here!?" Mermaidman: "You may kiss the bride!!!"
VeggieTales: The verses in "The Pirates Who Don't Do Anything" by Larry the Cucumber, who follows two pirate-themed verses with lines about never having kissed a chipmunk or painted daisies on a big red rubber ball. The other two go on to lampshade the hell out of it.
On Clerks: The Animated Series, while trapped in the freezer Randal sees an old lady shoplifting and he yells "The weeds of crime bear bitter fruit, you old hag!" We then see a flashback to the last time he said that, where it comes out of nowhere and makes no sense.