Irrational Fears are illogical fears that most demigods are sometimes plagued by. Demigod fears are often based on their parents' stories or sometimes in traumas they had when they were young. They are caused by Phobos, as shown in The Demigod Files.
The demigod children of Aphrodite are afraid of not looking perfect all the time, or they have a mild case of cacophobia (fear of ugliness). They are also afraid of not being fashionable. Piper McLean, however, is one of the extremely few exceptions.
The demigod children of Athena have arachnophobia, the fear of spiders. They fear spiders immensely because of the story of Athena and Arachne, with Arachne's children seeking revenge for their mother as a result. Annabeth's fear of spiders shows itself many times during both the series, ultimately culminating in The Mark of Athena, when she finally meets Arachne herself face to face. This fear of hers was so intense, that Percy just reminding her of it made Annabeth instantly snap out of the addictive aura generated by the Lotus Hotel and Casino. Annabeth's arachnophobia, however, may have been more acute than even that of her siblings, because of her childhood experience with them, and her encounter with Arachne herself.
The demigod children of Hephaestus have acrophobia, a fear of heights. It is because their father Hephaestus was pitched off Mount Olympus by Hera (who blames Zeus) as a baby. It is mentioned in The Ultimate Guide. This is strange, because Leo Valdez, a known son of Hephaestus, flew the Bronze Dragon, Festus. Also, Charles Beckendorf flew Blackjack in the beginning of The Last Olympian, with neither of them ever displaying signs of fear in the process.
The demigod children of Poseidon have claustrophobia, a fear of confined spaces. It is because the sea is so open, free, and wild. Percy mentioned his claustrophobia in The Lightning Thief, but it does not seem to bother him in The Battle of the Labyrinth. It could simply be that they become restless when confined, but are still functional and it only acts as a mild annoyance, or it may be a mistake of the author.
Satyrs are afraid of caves and underground locations. This is because they cannot tell if there are monsters nearby, as the underground smells like monsters to them at all times. Grover, however, was able to overcome this fear, having journeyed to both the Underworld and the Labyrinth.
Tyson is afraid of Grover (who, coincidentally, is scared of him), mainly due to him sneezing from the latter's goat hair. However, while in the Labyrinth, Tyson goes with Grover to help protect him, while they search for the lost god Pan. After this quest, the two seem to be on much better terms with each other, with Grover even cheering for Tyson after the latter helped defeat Typhon. Tyson is also very much terrified of everything related to death, such as the skeleton warriors aboard Clarisse's ship, and Nico di Angelo (initially).
Clarisse's greatest fear, as mentioned in The Demigod Files, is paterphobia, or more precisely, her father Ares threatening, scolding, and punishing her, as the god of war is infamously ferocious. However, with some encouragement from Percy, Clarisse was able to overcome it, and overpower Deimos.
Due to his life force being tied to a piece of fire wood, he has pyrophobia, causing him additional problems when dealing with Leo. However, after Leo provides him with a fireproof bag in The House of Hades, Frank's pyrophobia is almost completely overcome as a result. Frank also greatly fears Nico di Angelo (to the point that he doesn't dare to ask him personal questions and is terrified of going anywhere alone with him), even though he knows that the son of Hades is a loyal ally, as the brother of Frank's girlfriend. By the end of the series, however, Frank's fear of Nico seems to have greatly decreased in magnitude, as Nico approves of his relationship with Hazel, and Frank (as the new Praetor) makes it clear that Nico is always welcome to visit Camp Jupiter.
Grover Underwood has an intense fear of Cyclopes, fearing that he will be eaten by them. It can be traced back to when he was leading Luke, Thalia, and Annabeth to Camp Half-Blood, but took some wrong turns in Brooklyn and got trapped in a vicious Cyclops' lair. Then, there was the time Polyphemus was going to eat Grover, an horrifying ordeal that the latter barely escaped. Hence, Grover initially had problems with Tyson, and the feeling was mutual, but they both eventually got over this, with Tyson ultimately helping him find Pan. Grover also has an odd fear of rabbits, leporiphobia, which Thalia jokingly calls his "bunny-phobia." Grover's reasoning for it is because bunnies "steal carrots from defenseless satyrs."
After nearly dying in a muskeg in Alaska, Percy developed a fear of suffocation so intense, that it actually made him wary about traveling underwater, despite knowing that he can breath underwater. As The Mark of Athena progresses, Percy comes to believe that his fear is not just of physical suffocation, but of being "suffocated" by the burden of responsibility for his friends' safety which he places on himself. However, he does ultimately overcome this fear later, in The Son of Neptune.
Thalia Grace has acrophobia, a fear of heights, as shown in The Titan's Curse. This is ironic, because Thalia is the daughter of Zeus, the god of the sky, so she is rather ashamed of this fear of hers as a result.
Nico's greatest fear is of someone finding out about his homosexuality. Therefore, he never stayed at either camp for long and did not make connections of friendship with most people, so as to keep them from finding out, becoming more isolated than any other demigod. This fear about not being accepted due to being gay, most likely comes from Nico growing up in the 1940s, where "non-traditional" sexual orientations were not accepted. Hence, in The House of Hades, when Cupid forcibly makes Nico reveal his secret in front of Jason, the former is nearly reduced to tears. When Reyna accidentally learns about his secret in The Blood of Olympus, she, much to Nico's surprise and relief, doesn't judge him, and still accepts him as a friend just as he is (though she is still very surprised at this revelation), supporting him without hesitation. Reyna's unconditional acceptance and support was what most likely gave Nico the courage to finally come out to a very surprised Percy and Annabeth at the end of The Blood of Olympus, indicating that Nico has finally begun to overcome this immense fear and insecurity of his. And he succeeds in fully overcoming it , as he is in an official relationship with his boyfriend Will Solace in The Hidden Oracle.
Magnus suffers from lupophobia, fear of wolves. When Magnus was fourteen a pair of wolves attacked his apartment and killed his mother. Since then he's been terrified of them. He shied away from the "cleaning wolves" at the Hotel Valhalla and was terrified when he faced Fenris Wolf.