Ironic Fear

This is when a character has an overwhelming fear of something that someone of their kind just shouldn't be afraid of. For example, a bird who's afraid of heights or a mermaid who's afraid of water.

Often, they get an episode or subplot dedicated to resolving this fear. Almost always Played for Laughs, although there will often be a dramatic moment where they are forced to confront said fear.

May be part of a "What Do They Fear?" Episode. Compare Why Did It Have to Be Snakes?, in which the thing that the character fears pops up at an unexpected and inconvenient moment, rather than being something that constantly confronts them. If a character is afraid of something they are, they're a Boomerang Bigot. If they become the thing that they fear, it's a Karmic Transformation.

Anime and Manga

 * Tokyo Mew Mew features the hydrophobic character with water (and swimming) related powers variant.
 * Brook in One Piece is an undead skeleton who is afraid of ghosts and zombies.

Film

 * The Movie of The a Team explains B.A Baracus's fear of flying using this. After his first experience with Murdock's piloting skills, he was so scared he could not stand flying anymore, despite being in a occupation where jumping from a plane was an occasional must. He was not happy about it.
 * Blue of Rio is a macaw with a fear of flying.

Live Action TV

 * Monty Python's Flying Circus has a barber who hates hair, which explains why he only pretends to do his job. He never wanted to be a barber, anyway; he wanted to be a lumberjack!

Toys

 * Tamaru from Bionicle. He lives in a Tree-Top Town and wants to be a pilot, but is afraid of heights.

Western Animation

 * American Dragon: Jake Long features a hydrophobic mermaid.
 * Ruby Gloom has a bat who's afraid of heights.
 * Extreme Ghostbusters had a "What Do They Fear?" Episode with a ghost who could make phobias come to life. Eduardo's fear was of death: ironic both because he was taunting Kylie earlier claiming she was afraid of death, and because he interacts with ghosts on a daily basis, so he knows there's an afterlife.
 * The polar bears Muk and Luk in Balto are afraid to go in the water. They mention that it's the reason they are shunned by the other polar bears.
 * Fluttershy in My Little Pony Friendship Is Magic is afraid of... a lot of things, but the one that fits here is her fear of heights. She's a pegasus.
 * Early in Season 2, Stella of Winx Club was shown to be afraid of heights, despite being a fairy. Later, she seemed to overcome it.
 * Silverbolt of Transformers G1 is afraid of heights, despite being a giant robot capable of transforming into a jet and being the leader of the Autobot's main air squadron, the Aerialbots. Despite this, he proves to be a level-headed, extremely qualified leader who always does his duty.
 * It's specifically why Prime puts him in charge of the team. He says that the demands of the position will give him something to think about instead of his phobia. The fact that he has a sense of caution is also probably a plus, since the other four at that point were all extremely reckless.
 * Broadside of the same series gets it twice as bad. He's a triple changer with the vehicle modes of a jet and an aircraft carrier (yeah, yeah, we know), but he's both afraid of heights and gets the robotic equivalent of motion sickness at sea. Mocking his much unenviable lot in life is an infrequent joke among the fandom.

Video Games

 * Youmu from Touhou Project is a half-ghost who is afraid of, you guessed it, ghosts. Even more ironic in that her mistress is a ghost, and Youmu has never reacted to that.
 * Youmu is not so much scared of real ghosts as much as she is afraid of ghost stories.
 * Youmu isn't scared of much, but has trouble dealing with scary things. Imaginary horrors are her weakness: in one of the canon manga she fails a test of courage when she keeps fleeing hysterically from people in cheap halloween costumes, even though she was perfectly aware they were false.
 * Romeo, the sniper of the squad of Halo 3: ODST is implied to be afraid of heights.

Truth in Television

 * Many people in the health care field have stories of treating big, tough bikers with multiple tattoos and piercings who are worried that getting a shot or having a blood sample taken will hurt.
 * Ulysses S. Grant, a Civil War General, was so terrified of blood that he couldn't eat undercooked meat.