During the "Knightfall" saga of Batman, the mayor of Gotham is revealed to have this fear (as revealed by Scarecrow's fear toxin.) Joker, who is partnered up with Scarecrow, naturally exploits this for all it's worth by terrifying the mayor with a baby rattle (to remind him of rattlesnakes.)
Susanita from Mafalda is terrified of turtles. Unfortunately, her best friend Mafalda is unaware of her phobia and invites Susanita to her house just to show her the family's new pet turtle, Burocracia. Cue Susanita cowering in a corner.
Linus van Pelt of Peanuts is afraid of snakes. Or, more particularly, of queen snakes.
In an unpublished The Powerpuff Girls story, "Mojo's Day Off," Mojo Jojo proclaims he hates snakes. It freaks him out how they can move without legs.
In her third graphic novel, Sisters, Raina Telgemeier is revealed to be afraid of snakes. She wouldn't be afraid of them, had she paid attention to where she was stepping in that berry-infested forest! (She actually stepped on a snake, to be honest.)
Insects (entomophobia) and Spiders (arachnophobia)
Atomic Robo's fear of bugs is not irrational. Oh no. Fearing icky, evil things that can crawl into your body and die and get their sticky bug guts all over your insides and make you gross forever is completely logical!
In the first issue of Avengers: The Initiative, cocky newcomer Armory is revealed to be deathly afraid of spiders, to the point of accidentally killing one of her fellow trainees after Trauma involuntarily transforms into a giant spider.
In the Batman comics, the Scarecrow has developed chiroptophobia following years of defeats at the hands of Batman.
In the comics, at least for a while, Batman becomes the only thing he's afraid of.
Scarecrow, the Master of Fear and Lord of Despair, is sometimes depicted as having a crippling fear of birds, explained by some writer's description of creative abuse on his grandmother's part. Makes sense — who wouldn't want their alter ego to be the thing that scares the thing that scares them?
A Deadpool comic involving him trying to join the X-Men included him continuously pestering Domino to find out what's her greatest fear; turns out it's chickens. While this seems to be a Big-Lipped Alligator Moment for most of the arc, at the end where Domino and Wolverine are tracking Deadpool down they have to go through an air vent occupied by a chicken. Domino will probably Never Live It Down that a chicken took her out of action in a fight.
Flying (aviophobia) or Heights (acrophobia)
Cutter, chieftain of the Wolfriders and main character of the storyline, in ElfQuest. The revelation of this phobia is a significant plot point, midway through book one in the series: Cutter is required to 'fess up (from his subconscious, even) his worst fear specifically so he can be compelled to face and defeat it. Oh yeah, and there's also his rival Rayek and his fear of losing, but eh, who cares about him.
Bigby Wolf from Fables really doesn't like heights (since his ancestors weren't apes, he says).
Wolf in Marvel's Team America, but at least he can be kept conscious for a flight, although his powerful terrified death grip is hard on the seats.
Smudgy in Monica's Gang takes this fear to the extreme of not taking baths, which originates his nickname.
Difference Engine, a minor character introduced during Joss Whedon's run on Runaways was afraid of water, presumably due to his being a walking electrical appliance with primitive insulation.
Deadpool. He considers them the greatest threat to Earth, after Galactus. Amusingly, his solo mission in Marvel Ultimate Alliance had him fighting an army of clowns in a circus.
Popeye has an admitted fear of ghosts. (Or, "Evil spiriks" as he calls them.)
Tight Spaces (claustrophobia)
Storm, of the X-Men, is severely claustrophobic, the result of having been trapped under rubble after a botched aircraft attack that killed her parents.
Cloud Nine of the Initiative doesn't like tight spaces. Naturally, during Fear Itself, she was stuck under a pile of rubble, and proceeded to freak out.
The Post-Crisis incarnation of Supergirl spent three decades trapped inside a space pod in suspended animation. She gets frightened or mad when she's stuck inside a tight, dark space. In Girl Power she explains this to Raven.
Supergirl: You shouldn't have done that. Don't ever put me in the dark. Closed in. It's like I told Superboy. I can't handle it.
Toro of The 99 is claustrophobic. Luckily, his Noor stone allows him to teleport.
The Earth-2 Batman suffered an attack from the Scarecrow which caused him to hallucinate his greatest fear. It was fear of being alone, and he hallucinated his friends vanishing before his eyes.
Scarecrow did this to Batman in another story, but much more subtly. Batman didn't even realize he was affected, but found himself longing for a normal relationship to avoid spending his whole life along. As Bruce Wayne he begins dating, and almost marries, a woman who turns out to be a Black Widow.
In his Brazilian Comic book stories, Jose Carioca is so lazy that he's often depicted as having an outright phobia of work. That would maybe be "tripaliophobia".
In Empowered, the villain Rum, Sodomy, and the Lash is scared off by a fabric store. Apparently it brings up bad childhood memories.
In Geoff Johns' Teen Titans, Superboy and Beast Boy have a phobia of doctors, due to both of them being experimented on as kids. Beast Boy in particular has a special hatred for needles.
Lucy from Peanuts, in her role as a nickel psychiatrist, once diagnosed Charlie Brown as having pantophobia, "the fear of everything". He fervently agreed. This gag was repeated in A Charlie Brown Christmas.
Zatanna has a phobia regarding puppets, which means that her attempt to invoke "Sesame Street" Cred by guest starring on the show backfires horribly and ends with her throwing up in Oscar the Grouch's garbage can. It turns out, there is a very good reason for this phobia.
The Marvel villain Titania had a fear of Spider-Man, following her defeat in Secret Wars. It wasn't so much that he scared her as she felt powerless and humiliated, causing her phobia which lasted about a year (real time).
Klara Prast of the Runaways has a fear of corpses, presumably stemming from the time she was trapped in a sweatshop fire that killed most of the other workers. It's so severe that she once freaked out after Molly fainted during a battle, because she didn't know that that was part of Molly's powers and thought that she'd been killed. Needless to say, she does not react well when she gets stuck under Old Lace's corpse during an explosion...
New Avengers (2015): A henchman named Terry is forced to encounter both of his greatest fears when he and his boss encounter the just-reformed New Avengers, which is his case is squirrels and betrayal (his boss tries abandoning him to the superheroes).
Kitty Pryde, despite being an established fan of Star Wars and other science fiction films and having a life-long fascination with space, developed a severe phobia of space and the stars after being trapped in a phased missile flying blindly through space toward the end of Joss Whedon's run on Astonishing X-Men. One of the main reasons she fell in love with Peter "Starlord" Quill of the Guardians of the Galaxy is that he helped her overcome this fear.
The titular character in Big Nate has ailurophobia (fear of cats). This gets mocked by his peers whenever it's brought up.