Notable Non Sequitur

In a Detective Drama, any time a piece of dialogue comes along which is off-the-cuff, not followed up and unrelated to everything, you can tell it's going to be very important. If a suspect turns up late and says "Sorry I'm late, my car was stolen yesterday", the alleged car theft will be significant. If the detective remarks that the suspect has a nice keychain and the suspect says "Yeah, it's from my old fraternity", the insignia on the keychain will turn up later to reveal that the suspect and victim were in college together. Basically this happens whenever the writer can't find a neat way of dropping an important clue into an existing conversation.

See also Chekhov's Gun. Related to the Law of Conservation of Detail

Film

 * In the Olde-Tyme movie The Thin Man, early in the movie the odd watchchain of The Old Professor was pointed out. That same watchchain was later used in an effort to frame the Old Professor by leaving it at the scene of a murder. (But the Professor's alibi was solid -- at that point he had been dead for over a month.)
 * In Star Trek: First Contact, Geordi is giving routine orders to the engineering staff, and tells one of them to check the climate control, since it was getting a little warm in the engine room. It turns out that this was a clue that Borg had beamed aboard and were altering the ship, since Borg ships are warmer than Federation ships.
 * Hot Fuzz almost parodies this trope (like it does with most police movie tropes). Sgt. Angel makes one offhand comment about each of the minor characters before they are murdered that turns out to be the exact reason they were killed. The thing is, he had no idea. He came up with a complicated theory about why they were killed that had nothing to do with those offhand comments, and he found the killers before he was told their motive.
 * In Die Hard With a Vengeance, there is an offhand reports about thirteen dump trucks being stolen the night before. Later in the film, it is revealed that these trucks are being used by the villains to cart off the gold they have stolen.

Literature

 * In James Joyce's Ulysses, a man approaches Leopold Bloom, asking to read the report on the day's horseraces. Bloom, not wanting to be inconvenienced, tells the man to keep it, as he was just going 'to throw it away'. The man walks away, inspired. The winning horse of the day winds up being the horse, Throwaway.
 * In The Inheritance Cycle's first book, Murtagh tells Eragon his life story. The thing is, Eragon already knows the most important point--that --so the main point of the scene seems to be to drop enough hints for astute readers to figure out the next book's Twist Ending, that.
 * Any time Sherlock Holmes tells a client not to worry about going off-topic, this trope is in play. Notably lacking in the BBC modernization, where Sherlock has considerably less patience for the foibles of regular people.

Live Action TV

 * An episode of CSI had one character make an offhand comment about the goldfish in the pond on a suspect's property. Turns out they were pirahnas that the killer had fed the body to.
 * Castle uses this trope regularly, and inverts it almost as frequently. In almost every episode, minor details dismissed very early in the hour come back and provide a conclusive link to the true killer; in a number of episodes, similarly minor details pull the investigation in a different direction.
 * This happens early in Babylon 5 episode "Passing Through Gethsemane", twice even. There's a throwaway scene involving a news report about a criminal sentenced to mind-wipe, prompting Garibaldi and Delenn to briefly discuss its merits and flaws. Brother Edward idly mentions the story of Jesus knowing what would happen if he didn't leave Gethsemane, commenting that he doesn't know if he could do the same. later we learn Edward's been mind-wiped as a sentence, and while seeking forgiveness for his crimes he gets his chance to answer his earlier comment.

Video Games

 * There would be no Ace Attorney without this trope. Most of the contradictions Phoenix/Apollo/Edgeworth catches are off the cuff, seemingly unimportant statements that wind up screwing over the rest of the witness's claims.