Premonition Dreams are dreams that reveal a future event to the dreamer. Their contents vary widely, but each contains the Cosmic Owl. Croak dreams, a subset of premonition dreams, reveal how the dreamer may die. The dream may not be literal and is open to interpretation but will eventually happen.
In "The New Frontier," Jake has a croak dream about a Banana Man and a rocket ship in space, and suffocated because he did not have a space helmet. At the end of dream, the Cosmic Owl appears and wakes Jake up, which Jake claims confirms it was a croak dream. It is unknown whether one can cheat fate by avoiding a croak dream's circumstances, but Jake says it is impossible. It does seem possible, however, for one to indefinitely postpone one's fate, as Finn's presence postponed Jake's, since at the end of the episode, Jake states that the events in his dream will happen again when Finn is not around. They eventually did in "The Comet," however, it was revealed that the dream was not actually predicting his death, but was instead foreshadowing Banana Man saving Jake's life from either getting lost or suffocating in space.
In "The Lich," Finn has a premonition dream. In it, the Cosmic Owl appears on a TV that Finn is watching. At the end, the Lich leaves the snail he was possessing and kills Billy. When Finn and Jake arrive at Billy's cave, Billy appears unharmed, but it later reveals that the events of the dream had already happened by then and that the Lich is wearing Billy's skin.
In "Frost & Fire," Finn learns of his future breakup with Flame Princess when the Cosmic Owl tells him he "blew it" in his dream. Jake calls it a "prophetic dream," though the two terms are likely interchangeable.
The mere presence of the Cosmic Owl will make the dream come true as seen in "Hoots". If an outsider were to change a dream while the Cosmic Owl is present, the dream is destined to become true even if it was not the original intent. While uncommon it is possible to talk to the Cosmic Owl within a dream to ask for clarification. As seen previously it is possible to fall asleep quickly enough to follow the Cosmic Owl.
In the episode "Hoots," Finn, Gunter and Princess Bubblegum have dreams that become prophetic dreams. Finn dreams that he is in a purple room with Martin and Sweet P along with Jake taking photos and most importantly, the bird lady taking out a disco ball, some of it revealed in The Comet episode. Gunter dreams of a restaurant where humanoid penguins were having dinner and served food, and one waiter slipped and spilled all shrimp puffs. At the end of the episode, the dream events occur with Ice King's penguins. Princess Bubblegum's dream is disturbed by Gunter changing it so she is betrayed by the people of the Candy Kingdom. This comes true in "Hot Diggity Doom" when the Candy People choose the King of Ooo over Princess Bubblegum as their new leader.
In "Come Along With Me," the Cosmic Owl flies over Finn and Fern as they sit on the shore of Ooo in their shared dream which also made Fern's disappearance come true. Later, Finn and Jake sit on the same beach in real life.