Haruhi Suzumiya manages a mix of Agent Mulder and Agent Scully. While on the outside she wildly pursues all manner of fantasies, her inner common sense prevents her from realizing the fantastic elements right beside her. When Kyon tries to spell everything out to her, he gets shut down, because this would be "too easy". Itsuki later theorizes that deep down inside she just wants to have fun with friends.
Nurarihyon No Mago: Being the resident male-version Haruhi of the SOS Brigade Kiyo Cross Supernatural Squad, Kiyotsugu goes haywire at any possible sighting of youkai. Notably different from Haruhi in the fact that his obsession makes him more popular.
Maniwa from Paranoia Agent is the "Mulder" to his partner Ikari's "Scully", an open-minded, experimental detective who readily believes and accepts the true nature of the mysterious Shonen Bat while others deny it. Of course, this eventually turns him into an insane homeless person who thinks he's a superhero. The real Mulder never had to put up with that.
Durarara's Celty tends to buy into numerous paranormal and pseudo-scientific conspiracies (including The Greys. EspeciallyThe Greys), probably because her own existence suggests that they're entirely possible. There's even a chapter of the light novels devoted to her boyfriend assuring her that the world won't end in 2012, the dinosaurs weren't killed by The Greys, the photon belt will not destroy the world (nor is it sentient), and evil tesseracts will not come from the 4th dimension to conquer us all.
Peter Jackson gave us an example of this trope in The Frighteners. Special Agent Milton Dammers offers a view of what Mulder might be like if the next fifteen years were particularly unforgiving. Having spent almost two decades undercover in various cults without any therapy or deprogramming, Dammers is neurotic, paranoid, and absolutely insane. Perhaps not the best possible example since, though he knows something supernatural is going on, he jumps to more than a few wrong conclusions before catching on.
Conspiracy Brother in the movie Undercover Brother is one of these, though of the Dale Gribble variety: throughout the movie, he believes that he has uncovered numerous conspiracies by "The MAN," to eliminate/subjugate/neutralize/etc black people throughout the world, conspiracies that are dismissed by his coworkers and are indeed false. This is especially surprising considering the fact that there really is a shadowy white-supremacist world leader known only as "The MAN" dedicated to using a world conspiracy of powerful white people to keep down black people. The Dale Gribble part comes in regarding the crux of the movie: Not only does Conspiracy Brother miss the most legitimate conspiracy in the entire movie, putting brainwashing chemicals in fried chicken, but he actively furthers the scheme more than any of his coworkers who arrived at the truth long before he did.
Danny Bennett, from Tom Holt's early works, a BBC journalist with a wide range of conspiracy theories, all tying to the ultimate power behind world history: The British Milk Marketing Board. Although he is slightly vindicated when he gets caught up with Montalban and the Lombard Bank in Flying Dutch
The eponymous character of The Longing of Shiina Ryo is perfectly okay with the protagonist's unusual backstory, despite the fact it strongly establishes the world is filled with ancient conspiracies and bizarre supernatural entities.
Luna Lovegood of Harry Potter baffles her peers with her wacky theories and beliefs, on everything from mythical creatures with unlikely names (example: Crumple-Horned Snorkack) to conspiracy theories involving combinations of dark magic and gum disease. While she never proved the actual existence of any of these things, her quirky faith in people and creatures stood her in good stead through the series and she was shown to have uncanny insight. Her father, the editor of the wizarding tabloid, "The Quibbler", is a strong influence on her and he is much, much stranger. It is interesting to note that, since it is a magic/fantasy universe it is sometimes hard to tell why the other characters react with such disbelief towards her.
Epiny Burvelle, Nevare Burvelle's cousin in Robin Hobb's Soldier Son trilogy, will believe in just about anything supernatural and is often ridiculed for it. Since these are fantasy books, it should come as no surprise that she's often right, though.
Bailey School Kids: Liza is almost always the one to suggest that someone is a supernatural creature.
Although this was occasionally subverted, as in the season 7 episode "Hollywood AD":
Micah Hoffman: I am become Jesus Christ.
Mulder: I am become... skeptical.
In fact, whenever the Monster, Mystery or Victim of the Week was religious in nature, the "believer/skeptic" dynamic between Mulder (an agnostic atheist) and Scully (a devout Catholic) would be inverted, with Mulder becoming overly skeptical and Scully becoming the believer. Most references to Mulder and Scully seem to forget this including the latest movie.
To be fair, though, by the end of the series, Scully is much less of a skeptic and becomes almost Mulder's ally in the supernatural. In season 8, she is the believer in the X-Files office, while Doggett is the skeptic.
And while Mulder remains a steadfast agnostic through the series, Scully starts of a devout Catholic and degenerates over the course of the series. By "Redux II", she is far removed from her faith, breaking down when she realizes that she has tried everything except faith to help her cancer. She grabs the cross she wears and asks her mother why she wears it. With the advent of Bible verses being carved on the artifact in "Biogenesis" and her Buddhist/Christian experience in "all things", Scully is pretty conflicted about her faith.
And Scully has always had conflict between her faith and her science. In the episodes centered around religion, she finds it hard to reconcile what she knows of religion and what she knows is physically possible. From "All Souls":
Scully: As much as I have my faith, Father, I am a scientist trained to weigh evidence… but science only teaches us how... not why.
The earliest example of a reversal is the episode "Beyond the Sea," in which a convicted killer on death row claims he's a psychic and that he can help them solve a kidnapping case. Mulder is skeptical, believing the killer has orchestrated the kidnapping himself, but Scully gradually becomes convinced when he seems to be sending her messages from her recently dead father, such as the titular song lyrics.
Detective (or Sergeant, depending on the season) Munch from Law & Order: Special Victims Unit. He espouses conspiracy theories constantly throughout the series, usually for delivery of a moral point or for simple comic relief. Portrayed by Richard Belzer.
Agent Cooper from Twin Peaks was as accepting as Mulder (maybe even more so), but he lived in a much weirder world. Possibly the weirdest world ever portrayed on television. Which meant that he was almost always right! To the extent that he conducts successful murder investigations by throwing a bucket of rocks at a glass bottle.
John Locke from LOST is a good example of this, particularly when he believes he has to push a button every 108 minutes to save the world.
Daniel Jackson of Stargate SG-1 is more than a little used to having his theories laughed at, but the real kick is that even though he is right, national security ensures that he can't prove it to anyone.
Special Agent Paul Ballard from Dollhouse is very determined to expose the Dollhouse organization, and everyone else in the L.A. FBI office knows it and thinks he's a wacko.
Kate Lockley from Angel has an Agent Mulder attitude towards the paranormal world that Angel lives in. At one point, one of her fellow police officers compares her to Agent Scully, prompting Kate to try to explain why Mulder would be a better comparison. (This other officer knew little about The X-Files; he mentioned Scully because all he knew was that "Scully's the chick.")
Though she didn't become the Mulder until several personal experiences with the supernatural. Such as being attacked by a vampire.
Inspector Andy Trudeau is this in the first season of Charmed, with Inspector Morris as his Scully.
Hodgins in Bones falls under this category sometimes. Most characters are Scullys to him at one point or another.
An interesting twist in Psych, where Scully is Shawn, who pretends to be psychic, while Mulder is Gus, the "Non-Psychic" one.
Richard Castle, believing in psychics, ghosts, and in one episode where his status as Agent Mulder was explicitly stated, aliens. Beckett, of course, is his Agent Scully. Though, surprisingly Castle doesn't believe in curses while the rest of the squad is trying to convince him that they do. It turns out they were really screwing with him throughout the episode until he really does seem to be suffering from extremely bad luck after being attacked by a dog. Once this occurs the positions are reversed to normal.
In Undead Again he admits to Ryan that the reason he does this is merely to drive Beckett crazy, not that he actually believes it. Also as a writer he is mostly interested in what makes the best story as opposed to what is really occurring.
Marshall Teller believes in just about every conspiracy theory, urban legend and supposed cryptozoological siting he reads about. He even swears he saw Santa Claus. He doesn't have a Scully, but considering he lives in an entire town of oblivious skeptics, he doesn't really need one.
Marshall Eriksen from How I Met Your Mother believes in everything supernatural, examples include ghosts, miracles, aliens, Nessie and Sasquatch. He and his fiancée Lily once found a mutant animal they called 'cockamouse' as it was a combo of a cockroach and a mouse. Much like Agent Mulder's proofs of the paranormal, the cockamouse flew away.
666 Park Avenue: Detective Cooper seems to believe Jane when she says she was attacked by a ghost.
Constable George Crabtree is the one most likely to believe that vampires, ghosts, werewolves, Martians, Venusians, or an Egyptian curse might be responsible for the crimes that are being investigated. Detective Murdoch is the resident Agent Scully, Doctor Julia Ogden is of scientific mind as well, and Inspector Brackenreid gets angry easily by these superstitious ideas.
Doctor Emily Grace, who joined the series in season 5 believes in the afterlife and that it's possible to scientifically prove it. She also thinks that ghosts exist and she and Crabtree try to gain evidence.
Don Quixote is used in the Call Of Cthulhu roleplaying game to name two distinct psychic mechanisms (aka psychosis) to save either the character's life or sanity as an effect of overexposure to Cosmic Horror Stories: The Don Quixote, who sees the supernatural threat in every mundane object, and The Sancho Panza, who explains away every clearly supernatural event as just normal.
Persona 2 features two of these: Akinari Kashihara and Maya Okamura, both World History teachers at Seven Sisters High School. Kashihara wrote an entire book of conspiracy theories — ranging from Mayan prophecies to Nazi experiments to aliens living beneath Japan — based entirely on the ramblings of a mentally-disturbed student. Following his suspcious death, Okamura took up Kashihara's halberd, so to speak.
Tiffany Tiger's Panda roommate, Yin, in the webcomic The Suburban Jungle, fits this fairly well. She's a font of crackpot conspiracy theories, and yet she manages to become a prisoner of 'The Village', abducted by aliens, and gains help from a dimension-hopping polar bear...
Invader Zim is rife with Agent Mulders, ranging from the Swollen Eyeball organization who look to prove that mysteries and conspiracies are true, to clueless FBI crackpots (Count Cocoa Fang the cereal mascot being a real vampire) to disturbingly on-the-ball lice hunting countesses (there's a giant Alien-like lice queen who's responsible for lice infestations).
On the other hand Dib has claimed to see Big Foot in his garage ("he was using the belt sander") and a vampire hamster, and we never did find out if that was a baby sasquatch or a hairy kid — although its comment about wanting to eat grubs seems to suggest the former. So, depending on the episode, Dib tends to be either the Mulder or the Scully.
Mr. Crocker of The Fairly OddParents is known for muttering insane theories to himself that climax in him doing a wild take and yelling "FAIRY GODPARENTS!" in a shrill voice at the top of his lungs. The scary part is that he's almost always right, and he still comes off as a raving loon to the viewer.
The viewer may be smarter than you think: while he is usually right, only once in a blue moon (notably in the movie, when he somehow manages to activate a wish-granting muffin) does he get any evidence to support his ideas (and when he does it's promptly magically wiped out of his head), the vast majority of his fairy-chasing plans not only come to naught but result in severe punishment when he accidentally does unpleasant things to his employer, and he just keeps at it. And then there's the yelling (generally accompanied by anatomically frightening poses)...
Interestingly, in one of the Time Travel Episodes, we learn that Cosmo and Wanda were his fairy godparents before they were Timmy's (they had surprisingly similar childhoods, including Vic, the babysitter) and he escaped the Mind Wipe of having them go away forever when they were revealed by writing a note to himself on the back of a device with which Timmy provided him through time travel, so he's not a crackpot after all.
Captain Black of Jackie Chan Adventures is the show's Agent Scully for most of the first season - until he gets an up-close look at Shendu's true form in the season finale. He switches to Mulder for the rest of the show.
Dale Gribble of King of the Hill is a conspiracy theorist and generally a survivalist nutcase. He uses his alias Rusty Shackleford for everything, even ordering pizza. He is so devoted to conspiracy that figuring out the flaws in the "second shooter" theory of the Kennedy assassination turned his entire life around, turning him into a pro-government "good citizen" until Hank, against his better judgment, convinced him that not every conspiracy theory is necessarily false. Fittingly, his design was based on a combination of Robert Patrick and a composite sketch (mostly the horribly inaccurate one of the Unabomber).
The Question in Justice League Unlimited. The guy believes that aglets have a sinister purpose, and that topically applied fluoride renders teeth visible by spy satellite. On the other hand, he's proven to be a highly talented detective, either in spite of his paranoia or even perhaps because of it. (And he did nail it with his theory about Baskin Robbins' secret, hidden Flavor 32.)
Martin Mystery really likes crazy theories about aliens and so forth, based mostly on comic books and B-movies. However, the fact that he and his sister, Dianna work for an organization dedicated to fighting aliens and so forth makes the theories much less crazy — well, some of them. He goes through a lot of downright ludicrous explanations before he finally arrives at the one that's actually happening.