At the end of Monster, Tenma has a dream (or not) of Johan revealing to him what it was about his childhood that plagued him the most.
The second episode of the anime; Fujiko left Lupin a few cryptic negatives with strange lines and symbols that belong to Pycal/Piker, a menacing magician. Lupin had already figured out his fire-shooting and levitation tricks but is completely clueless about how he becomes invulnerable. Later, however, Lupin accidentally falls out the window and the resulting blow knocks him into some sort of dream where the negatives are projected over each other, revealing a chemical formula for an extremely resistant liquid.
In Tintin in Tibet, Tintin has a dream about his friend Chang surviving a plane crash in the Himalayas. The next morning he hears about the crash in the paper and goes to search for Chang, taking the dream as a sign.
In K.A. Applegate's Animorphs, it turns out #41 was all a mind exercise, during which Jake rediscovers the things they're fighting for in the first place.
In Madeleine L'Engle's The Other Side of the Sun, Stella (an Englishwoman) goes to live with her husband's father in the Deep South shortly after the The American Civil War. After a dream involving fireflies and her husband metamorphizing into a (black) man she had met there, she wakes to the realization that her husband and this man are half-brothers.
Used in Twilight after Bella learns Edward is a vampire. Even though Dream-Edward is attempting to lead her to her doom, Bella realizes she doesn't care if Edward's a vampire as she's more upset that Dream-Edward is about to be attacked. The same dream with a slight twist is used in the sequel New Moon so she can realize Jacob is a werewolf.
In Graham McNeill's Warhammer 40,000Ultramarines novel The Killing Ground, the Lord of the Unfleshed remembers deliberately putting himself somewhere where fumes would make him dream deeply, and remember his past.
Inverted in The Second Chronicles of Amber. In Sign of Chaos, Merlin muses (paraphrased) "Someone who had been through all the truly bizarre crap I had over the past few days should have had a revelatory dream, waking up with new insights as to how to deal with their problems. Me? I woke up in the middle of the night and realized that my feet hurt."
Buttercup has dreams in The Princess Bride which upset her enough to make her want to call off her wedding to Prince Humperdinck. The film gives her just the one, but in the book she has bad dreams for a good week.
In Felidae, Francis the cat has a highly disturbing dream that symbolises the motive behind the murders he's investigating. He still has to do some more conventional investigation to work out what the dream means, though.
Menelaus, owing to his mental modifications, can consciously think while dreaming—which still leads to this.
In the second book, Mickey uses drugs and dreams that the Judge of Ages is awake and among them. The Witches explain that it pieces together insights. Menelaus, who is the Judge of Ages awake and among them, finds it awkward.
In Barbara Hambly's Ran Away, Benjamin January dreams of his dead wife asking where Sabid is — which causes him to consider whether Sabid might actually be in New Orleans, making trouble again for the same man he attacked years ago.
In Thief of Time Jeremy clockson has this happen while working on the glass clock. He wakes up to find his sheets and wall covered in diagrams and notes.
In Michael Flynn's Up Jim River, Donovan, in his drugged sleep, realizes a few things (on top of getting his soul pieced back together).
In the first Harry Potter book Harry Potter and the Philosopher's Stone Harry has a dream about wearing Professor Quirrell's turban and hearing a voice coming from it. He wakes up and... promptly forgets it. If he hadn't ignored this dream then he could have figured out the truth way before he did.
In Poul Anderson's "A World Called Maanerek", Wanen dreams of his shipmates dying painfully. Horlam explains that it is merely the revelation of his impulses toward destruction: that their animal nature does not fit well in the Hegemony, and that a good unit does not deny it but is strong enough to overcome it.
In George MacDonald's "Port In A Storm", the narrator explains to his children that he had known of the winecellar as a child, and dreamed of it the night before he found it again.
In Patricia C. Wrede's Frontier Magic novel The Far West, Eff eventually realizes that the vivid dreams she has been having are a result of her magically nudging herself to pay attention to them.
In Andre Norton's The Zero Stone, a feverish Jern dreams of hostility from the medico of the ship — turns out they had taken him to hand over to his enemies.
This is the likeliest explanation (it's left ambiguous) for Russell's visions of the Four Heroes in You, as immediately after most of them he mentions waking up.
The Incredibles originally was to include a Dream Sequence where Helen's worries about Bob took shape. This was one of the first things cut from the plot partly because it was too lazy. It's discussed on the DVD extras, though.
The Dark Knight Rises: Bruce sees Ra's al Ghul appear before him in a hallucination, and in the ensuing conversation he comes to think that Bane is his son. He's not, but by tracking him he eventually discovers who is Ra's al Ghul's daughter.
Twin Peaks—Agent Cooper has an iconic dream early on in which the victim whispers the name of her killer. He immediately forgets it, however, and it's a while before he is able to remember.
In one episode, the not-so-good doctor had a dream which gave him the key to resolving his latest case. The dream also involved him wearing a catheter which ruptured, but that had more to do with his own issues at the time.
In another case, they used Dreaming The Truth, but the audience can't be sure if he just hadn't put the facts together or if he was actively in denial. After all, who wants to remember that they got their best and only friend's girlfriend in a bus accident?
Tony had a dream where he was forced to face that he knew one of his underlings was wearing a wire.
In the same series, Dr. Melfi has a dream which leads her to realize that she can, if she wants to, have Tony punish her rapist - though she chooses not to.
And in Season 6, Carmela has a dream where she's forced to confront the possibility that Adriana was killed. Previously, she'd been led to believe that the latter had abandoned the family.
Battlestar Galactica: Whilst near death, Laura Roslin hallucinates and dredges up a forgotten memory of seeing Gaius Baltar kissing a Number 6 on Caprica before the attack. Though she met both later, she didn't connect the two in her head until that moment.
This happened in an episode of Medium. A Texas Ranger was troubled by dreams that had only recently started to become clear... and then the things he was dreaming started happening (a pair of EMTs killed and dumped where he dreamed they would be, and so on). It later turned out that he truly wasn't having prophetic dreams; he'd spent some time comatose in a hospital, and while in that state had overheard someone making plans that later presented themselves as dreams as his mind made sense of the information.
2-parter "Human Nature/The Family of Blood" has the temporarily-human/amnesiac Doctor/John Smith dreaming about his Time Lord adventures and recording them in his Journal of Impossible Things.
In the Series Five episode "Amy's Choice", the Eleventh Doctor's feelings are revealed when psychic pollen gets stuck in the TARDIS time rotor, heats up, and induces a dream state for Amy, Rory, and the Doctor. The pollen latches on to the Doctor's massive amounts of darkness and reveals the difficult truths about Eleven's character to Amy and Rory in two dreams.
Walter Sherman has the occasional Dream Sequence which ends in him figuring out something he missed earlier in whatever case he's working. Usually its about the nature of what he's looking for or an aspect of a person involved in the case that he misread. Once, instead of a standard dream sequence, he hijacked a hypnotic state and used it to figure out a mistake made in a 20 year old murder case.
Used on Psych in the "Mr. Yin Presents" episode. Shawn is able to remember a vital clue that he walked right past the killer in the movie theater the night before and saw he was wearing ankle weights. Subverted by the fact that the ankle weights clue was a Red Herring and the one Shawn thought was the killer is actually Yin's next victim.
In Jays Journey, Gaia has a dream where everyone in the dream (save Gaia herself) repeats something said earlier in the game, which lets Gaia see how selfish she's been, and why Carol's been acting so cold to the man she previously loved. And to get an opportunity to repeat all the game's Running Gags in quick succession.
In Um Jammer Lammy, guitar-dependent Lammy has a Dream Sequence about a famous TV martial arts master being the new lead singer of her band. They play a rather prophetic song together that slowly derails into Shave and a Haircut, and Lammy realises her guitar is actually a vaccuum cleaner. The Sensei then tells her that he lost his dojo, which even had a casino, but it's okay, as it still exists in his mind. This gives Lammy the inspiration she needs to realise her guitar is just a Magic Feather, and she can be badass without it, since it's in her mind.
In Westward, an Intrepid Reporter who was once captured and tortured by communists starts having strange dreams about the experience many years later that don't match his conscious memories. Eventually he confirms that his dreams reflect reality, and this leads him to stumble upon a mystery of potentially cosmic proportions.
Happened a few times on The Simpsons. Twin Peaks was even explicitly parodied at one point during one of these.
In one early Family Guy episode, Lois dreams the truth; that Stewie's an evil baby genius hellbent on world domination. Of course, when she wakes up, she immediately forgets the details and brushes it off as a bad dream.
The practice of "sleeping on a problem" is common in any field where mental roadblocks are commonplace. Often, forcing yourself to sleep will at least help you be more refreshed and ready to tackle the problem later; also, when you're asleep, your subconscious mind is free to make wilder leaps of logic without being bothered by the part of your brain that determines whether or not a particular idea makes sense.
The German chemist Friedrich August Kekule was trying to figure out the structure of the compound benzene. He had a daydream about a snake biting its own tail. When he woke up, he realized that the benzene molecule was in the shape of a ring, and this was later confirmed. The Other Wiki's take on it, and Mr. Kekule's own description of the experience.
Double Subverted by Otto Loewi, a German pharmacologist who earned the Nobel prize for his discovery of of the neurotransmitter acetylcholine. One night in the spring of 1921, he awoke with the sudden realization that he knew how the the brain's electrical signals traveled along synapses. He hurriedly wrote down his revelation and went back to sleep, only to discover in the morning that, to his horror, he could not read what he had written down. Fortunately, he had the same dream the following night. Taking no chances this time, he got up and went right to the lab where his experiments confirmed his idea.
There is a legend that Dmitri Mendeleev had seen the periodic table of the elements in a dream.
In George Orwell's "My Country, Right Or Left", he recounts how he had argued against the obviously approaching war (and all the people who would line up behind the government if it started) — but the night before the Russo-German pact, he had had a dream that war had started and it revealed to him that he, too, would support the war out of patriotism.
Maria Gaetana Agnesi, an Italian mathematician, solved some math problems in her sleep.