Ai/Shiho from Detective Conan. Once she had a dream about Gin knowing her secret identity and watching her and her friends from his car, planning to kill them. Later, when she and Conan find Gins car, Ai is terrified.
In Composure, Princess Celestia has dreams in which her sister becomes Nightmare Moon again and attacks her. It later leads to a mild misunderstanding when she wakes up in a hospital with lightning burns and Laser-Guided Amnesia.
In Eleanor Cameron's The Court of the Stone Children, Nina dreams of trying to look up her phone number in the book although they have just moved there and wouldn't be in it. (She's already gotten lost once.)
In L. M. Montgomery's The Blue Castle, Valancy dreams that she is confronting someone whom she told something that proved to be false, and he proves to be made of glass and breaks.
In Devon Monk's Dead Iron, LeFel does not sleep because he will only dream of dying, which is near.
In Devon Monk's Allie Beckstrom book Magic to the Bone, Allie comments on how she does not have them, sleeping on the drive to a friend's, escaping a city where she's suspected of murdering her father.
In Wen Spencer's Tinker, she suffers them regularly, a mix of current events tossed up with a dream maze.
Also by Wen Spencer, in A Brother's Price the Princess Rennsaeler has been having Bad Dreams about the theater explosion that killed her husband and older sisters. When she falls in love with Jerin, she starts dreaming that he's there too.
In Homer's Iliad, he uses an anxiety dream of being unable to flee when chased, or unable to pursue when chasing, as a metaphor for Achilles's chase after Hector, where Hector couldn't get away, and Achilles could not catch him.
In Patricia A. McKillip's "The Kelpie" Wilding suggests that Emma is sleeping poorly because of these; a future of matrimony and children will give her little chance to paint.
In Seanan McGuire's October Daye novel One Salt Sea, Anceline offers to take Toby swimming fast. Toby says she looks forward to that — in her nightmares, she thinks.
In Patricia A. McKillip's The Riddle-Master of Hed, Raederle is promised in marriage to whoever won the crown. After they learn it was won, and before they learn who it was, she confesses to her brother that she is having dreams of some unearthly bridegroom.
One late episode of Frasier is dedicated to the main characters' anxiety dreams — Frasier dreams that Roz is berating him because his radio booth is full of cobwebs and no one has called in six months, reflecting his worries that his show might be declining as well as his fear of being alone, Niles dreams about accidentally baking his unborn baby into a pie, reflecting his fear that he's going to be a bad father, and Daphne dreams about inflating comically while Niles cheats on her with several supermodels, reflecting her fear that motherhood and age will affect her marriage. Martin, on the other hand, just dreams about singing and dancing with his new girlfriend in the style of Fred Astaire and Ginger Rogers — without his cane.
In the first series of Buffy the Vampire Slayer, everyone's dreams start to come true. Most are "classic" anxiety dreams such as being in class in one's underpants (Xander), having to sing in public (Willow), losing the ability to read (Giles), and having to sit a test for which she had not taken the class or studied (Buffy).
In "Restless" Willow dreams that she's turned up for a play without having learnt the lines, then she has to deliver a talk in her Season One nerd persona before a classroom full of her jeering friends. Xander's dream involves him going to the toilet, only to find the entire Initiative is watching him and taking notes.
A more subtle version is in "Hush" when Buffy has an Erotic Dream of kissing Love Interest Riley Finn — except it's taking place in front of an entire class, showing her nervousness about trying to get his First Kiss.
In Flight of the Conchords, Mel has a song about wanting real life to be more like her dreams. Some of the dreams she mentions are stock anxiety dreams, like being naked in public and taking an exam you don't know any of the answers to. (Oddly, this doesn't seem to affect her desire to live in a dream world at all.)
In Rose Is Rose, the greedy Clem has a nightmare where he shared all the brownies and left none for himself.
In Peanuts, Snoopy dreams of Charlie Brown's flying him like a kite, so that he crashes into the earth and shatters. Waking he blames the 30-inch pizza eaten just before bed.
In Prickly City, the voter who cast the deciding vote to elect Kevin, Lost Bunny of the Apocalypse, to the Senate (and get him out of Prickly City), has dreams of his campaign for the presidency — combining this with Bad Dreams.