Shinigami in Death Note can only be seen by people who have touched a Death Note belonging to that particular shinigami.
In Blue Drop, Mari is the only one who can see the holographic projection of Tsubael, the bridge bunny from Hagino's spaceship, Blue. It is hinted that Michiko can see her as well.
In Fight Ippatsu! Juuden-chan!!, Charger Girls use a cloaking system to hide themselves from the populace of Earth (normally, they're transparent, but can still be seen), and while cloaked, can only be seen or touched by other people from Life Core. And Sento, for some reason. Sento's ability to see and touch them is never explained in the anime, but he does briefly lose the ability to do so while depressed.
In The Frighteners, those who have a traumatic experience sometimes have their perception altered. This allows them to see ghosts, the white light/stairway to heaven, and other... things.
In the James Stewart movie Harvey, Harvey is a pooka, a fairy spirit who appears to Stewart's character as a six-foot white rabbit, but who is invisible to others. Naturally, everyone else thinks Stewart is insane. One character bothers (named Wilson) to look up "pooka" in a dictionary and reads the definition, "a fairy spirit in animal form. ... very fond of rumpots, crackpots, and how are you today Mr. Wilson?" He's very upset.
In Beetlejuice, only Lydia can see the Maitlands. According to the "Handbook of the Deceased", this is because ghosts cannot be seen by the living because they are "strange and unusual", something that Lydia herself is.
They Live!!: The Aliens can only be seen for what they are by people with the special sunglasses (later contact lenses).
Thestrals in Harry Potter can only be seen by those who have witnessed someone die. Harry joins this group in the fifth book after he comes to terms with being there at Cedric's murder. It comes as a bit of a shock when he arrives at Hogwarts and sees that the "horseless" carriages are actually pulled by skeletal, reptile-faced, winged horses.
Death can be seen by animals, witches and wizards, and small children. And people about to die or who are having a near-death experience. In brief, anybody whose Weirdness Censor is off. ...And Carrot, who used him as a witness for a murder in the short story 'Theatre of Cruelty'. The book Wyrd Sisters states that Death can also be seen if people truly expect to see him (which may explain Carrot's ability to use him as a witness). In Wyrd Sisters, Death walks on stage during a play at a time when the audience expects an actor playing Death to show up — and is seen by everyone in the audience. And gets stage fright.
Discworld also brings us the expression 'First Sight' from the Tiffany Aching books. Apparently Second Sight is the ability to see things that aren't really there (most people have this), but First Sight is the ability to see things that are, whether or not you're expecting them. Wizards are trained to have this ability basically as soon as they start at the Unseen University, and it's a great help to naturally have it if you're going to be a witch. Carrot's extremely straightforward nature may correlate somehow.
Ghosts can, according to Death, be seen by close relatives, the magically-receptive, and cats.
The Bogles (weird humanoid demon-animals that inhabit the Underworld) from Wintersmith can only be seen by people who have their eyes shut at the time. This might be a deliberate parody of the Blind Seer trope.
The Ten O'Clock People, by Stephen King, had monsters that appeared as human but could only be identified by people who smoke a certain amount (roughly a couple a day, but not heavily).
It's based on a swindle, which in turn is based on the natural human fear of looking stupid. Hence, everyone who looks at the "magic cloth" pretends to see it, based on the "rules" that the con-men have given to the Emperor, for fear of losing their jobs. The "innocent" child who sees through the scam does so because he (or she) has not been trained to tell the Emperor and those around him what they want to hear (i.e., to lie in order to preserve the feelings of the powerful, and the life of the speaker). And the story itself is a metaphor for how a powerful idiot can only be challenged by someone who has nothing to lose.
One Whole Plot Reference makes it ambiguous whether it's a con... or whether the cloth really does work and the reason the viewpoint character could never see it is that he's indeed foolish and unfit for his position.
The classic short story Eight O'Clock in the Morning had monsters who could only be seen after awakening from hypnosis. It was adapted into the So Bad, It's Good movie They Live, where the monsters can only be seen by wearing special sunglasses.
In the novel Kit's Wilderness, the main character gains the ability to see ghosts after suffering a mysterious near death experience, but a few other major characters of all ages claim to see them as well, but there is no way of knowing whether they are telling the truth or not, and his closest friends seem to know when he is looking at them or in their presence.
Central to Theodore Sturgeon's classic novella, "The [Widget], The [Wadget], and Boff".
In James Clemens' novel Shadowfall and sequels, one character, due to her unique heritage, has a companion which only she can see or touch. (It can be made visible and tangible to others by her blood.)
In the Eoin Colfer novel The Supernaturalist, the Parasites can only be seen by those who have experienced a near-death experience, and the ability to keep seeing them is sustained by the excess of chemicals in the air. However, Ditto can see the Parasites because he is a Bartoli baby and also notes at the end of the book that there are other things out there many people can't see.
The Spectres in His Dark Materials can only be seen by children, but feed on the minds/souls of adults.
In Gustave Le Rouge's La Guerre des Vampires, Martians can only be seen by blind people (or by someone who's wearing a special Martian helmet).
In the Star Wars novel Dark Apprentice, Luke Skywalker's Force ghost can be seen only by the Solo twins, Jacen and Jaina. Although they see him through the Force, the trope still applies since the other Jedi were unable to detect him.
In Myth-ion Improbable, the lurking dust bunnies of the Vortex dimensions are invisible to guys. Of course.
How the Animorphs learn about the existence of the Chee, a race of androids living among humans through the use of holograms: in dog morph, Marco notes that Eric has no smell (though he was carrying smells picked up from touching other people), but because dogs have mammalian eyes, he still looked normal. It takes a spider morph, with completely different eyes, to reveal the machine under the hologram.
Inverted in Fredric Brown's Martians Go Home. The Martians are real, and everyone on earth can see them — in fact, it's impossible not to — except for one guy, thanks to a Martian-caused nervous breakdown.
In The Last Unicorn, normal humans can't see the titular unicorn (or unicorns in general). All they will see is a white mare at best. It is implied that only some humans who deeply believe in their existence, such as Molly, or being possessing magic, such as Schmendrick and the witch who captures her early in the movie, can see her who what she truly is. Said witch had to create on her a fake horn in order to make people believe she was a true unicorn, since they can't see her real horn. Haggard himself can see through her human disguise because, as he puts it, everything in her behaviour betrays her real nature. Finally, the skull saw right through her disguise.
Alan Dean Foster's Into The Out Of features three characters who are the only ones who can see the supernatural creatures infiltrating the world in order to destroy it. For two of them, it's a literal case of By the Eyes of the Blind, as both of them have lost an eye and can see the creatures with their glass eye. The third has a Third Eye.'
Faeries in The Spiderwick Chronicles can only be seen by people with the Sight. If you don't have it, you can get it by rubbing faerie ointment, faerie bathing water, or hobgoblin saliva in your eyes, or by carrying a four-leaf clover or a seeing stone.
The insect-like creatures from a parallel universe in Stargate SG-1 could only be seen by those irradiated by the alien device brought by SG-1, or those who touched them. Interestingly, the creatures themselves did absolutely nothing (since they had always been there), but suddenly being able to see them led to some... panic.
Also, beings with Naquadah in their blood (Goa'uld symbiotes and by extension anyone who has ever had a symbiote in their body) can sense when others have Naquadah in their systems. After Carter is briefly taken as host by a Tok'ra, her ability to sense Goa'uld comes in handy. Goa'uld can also sense Reetou, an insectoid race that exists out of phase from our universe.
Dawn's true nature can only be seen by mentally unstable people. Mentally ill or psychic people can see the energy of the Key that's hidden inside her.
The child-killing demon Der Kindestod from the Season 2 episode "Killed by Death" can only be seen by young people with fevers.
On LOST, Hurley, who is Genre Savvy, theorizes that only he, Locke, and Ben can see Jacob's cabin because they are the craziest ones.
In the final season of Slings and Arrows, Geoffrey and his ghostly mentor Oliver Wells hazard that the only ones who can see Oliver (besides Geoffrey...who is is probably mad) are people near death.
In Kamen Rider Dragon Knight, mirror monsters can only be seen by Riders and those few Muggles who have been to the mirror world. The same goes for being able to look into a reflection and see battles taking place in mirror world. When a fight takes place in reality, most people tend to see Riders punching and yelling at thin air.
Vorlons in Babylon 5 cannot be seen by those who have been touched by the Shadows.
In Angel only people whose blood has mixed with Jasmine's can see beyond her beauty to see the creepy maggot-infested walking Eldritch Abomination she is. Subverted with Connor who always saw her as a maggot-infested walking Eldritch Abomination because he was her father. Yeah. Talk about a face only a parent could love.
Al, the hologram sidekick from Quantum Leap could only be seen by Sam, animals and 'the very old, very young and very crazy.'
The X-Files episode "Folie a Deux" featured a monster disguised as a human working in a telemarketing company, escaping detection by apparently "hiding in the light", and is able to kill people and control their corpses without anyone else noticing they're dead. An employee named Lambert, however, is able to see him as he is, as well as those whom he killed. Everyone thinks he's crazy, but he manages to get Mulder to see the monster as well.
In the Doctor Who episode "Vincent and the Doctor", only Vincent van Gogh can see the Krafayis. No explanation is given for this, but it has something to do with his artistic mind and / or depression.
In the original series episode "Behold, Eck!", a two-dimensional creature from Another Dimension accidentally enters our world. It can only be perceived by people wearing glasses made from a special prescription that uses meteroic quartz.
In the revival episode "Music of the Spheres", the alien audio signal is only recognizable as music to teenagers, but not to adults or younger children.
In Sesame Street, only Big Bird could see Aloysius Snuffleupagus until 1985.
In Teen Angel, Grandpa Jerry can see angels and Rod because of a near-death experience.
In Grimm, the many types of monsters that inhabit our world, as seen in certain fairy tales, appear to be normal humans to everyone except other monsters and members of the Grimm family.
In Being Human, only people who are somehow supernatural (including vampires, werewolves, and zombies) can see ghosts.
Played with in the Star Trek: The Next Generation episode "Imaginary Friend". Young Clara originally has a not-unusual imaginary friend named Isabelle who she pretends to play with. Then an alien presense reads her mind and manifests itself as the physical embodiment of Isabelle. It's not really that no one else can see her though, just that she vanishes whenever someone else enters the room.
In the Gorillaz song "Fire Coming Out of the Monkey's Head", the Happy Folk were unable to see the Strange Folk, because the Strange Folk wore dark glasses, hiding their eyes. 'Without the truth of the eyes, the Happy Folk were blind.'
Billy Joel uses this trope word-for-word as a lyric in his song "River of Dreams".
In the music video for Lordi's "Blood Red Sandman", the monsters/band can only be seen at first through a handheld camera.
In the tabletop role-playing game Changeling: The Lost, a Changeling's mien (that is, their true, Fae-inspired appearance) is virtually impossible to detect against a Changeling's will, even to other magical folk; though there may be enough bleedover to make their mundane appearance unique or unusual enough to make others suspect there could be something wrong , the illusion of a normal visage, called the Mask, is complete and almost impenetrable. However, one book lists optional conditions for people who are naturally capable of seeing through it— such as the seventh son of a seventh son, a person with heterochromia, or someone born on Leap Day.
The same book has a sample character who can see through the Mask - when he was a kid, he took a bullet to the head, and it hit him in just the right way that his perceptions were opened for good. And it scares the shit out of him.
Then there's the fact that most things from the Hedge, such as hobgoblins, can only be perceived on Earth by changelings or any humans a changeling has chosen to ensorcel.
In the 3E Ravenloft supplement Dark Tales & Disturbing Legends, many bogeymen can only be seen by children.
Amaterasu from Ōkami is visible to everyone, but the Facial Markings that indicate her divinity only show up to people who still have faith in the gods. Similarly, she can only be understood by supernatural creatures such as Poncles, Celestials, and other gods. One particularily devout follower actually does happen to be blind, and all he sees is a beautiful maiden standing before him. Issun (a Poncle) is confused about what the guy is talking about, because he doesn't see anyone but the Furball herself.
In Doctor Lautrec, only Sophie can see the spirits in Treasure Animatus.
The gods of A Moment Of Peace are invisible to the stick figure humans they walk among, but apparently Monster Food, limbless worm-like creatures hanging from the very bottom of the food chain, can see them. Or at least this one can.
Monsterhood has an interesting variation. The monster town is contained in an energy field that makes those outside it believe that whatever is inside is fantasy — they'll rationalize their experiences within as a dream, a movie they were acting in, or anything else, and monsters trapped outside too long will even forget they are monsters, as monsters exist inside the field and are thus fantasy. The "Blind" here? People who can't distinguish between fantasy and reality are entirely unaffected, as what's within the field doesn't seem any less "real" to them.
The world accessible through SCP-093 is almost immediately found to be full of strange figures that are visible on the monitoring cameras, but which the people exploring the world can't seem to see. They turn out to be the "ghosts" of the Unclean and those absorbed by them — they can only be seen by machines or those affected by His Tears. They show no ability to speak, but seem able to communicate textually through machinery that allows it, and communicate telepathically with the affected.
SCP-870 can only be seen by schizophrenics, which causes everyone but the Foundation to dismiss them as hallucinations. It doesn't help that they have biologically impossible appearances which one would expect to only see in nightmares.
Implied in many works from The Slender Man Mythos. Typically, only Slender Man's intended victims can actually see him unassisted. However, he always shows up in pictures and videos, which is why the more Genre Savvy characters (i.e. Alex Kralie, Milo,) start keeping a camera with them at all times. Then, when other characters review said pictures and videos, they end up going "Hey wait, why didn't I ever notice that rather obvious tall, faceless guy in the business suit when I was actually there?" Unfortunately, realizing that Slender Man exists is an easy way to get on his list...
In an episode of Skunk Fu! the invisible ninja monkeys could only be seen by stupid people so Ox the dumbest cast member was the only one able to see and fight them. Skunk and the others think of stupid thoughts and are then able to see them.
In the Sushi Pack episode "The Thing That Wasn't There," only Maguro can see the electrical monster that's wreaking havoc in Wharf City. Whether or not this is due to her mental powers is never explained, and she herself says she doesn't know why she can see it while the others can't.
Spoofed in The Simpsons with Osmodiar, a similar-looking alien that can only be seen by Homer. And indeed, the audience can't see it either.
One episode of Danny Phantom was centered around Danny childishly breaking stuff and blaming a ghost that no one else could see. Turned out, he was right; the ghost was only visible to children or those with a similar mentality. Jazz was eventually able to see him once she was forcibly reminded she's not as adult as she likes to think.
Olfactory example: There was a study about how asparagus imparts a particular smell to urine, yet some people report that it doesn't work on them. The punchline of that study was that some people simply cannot detect that particular smell - they have a "dead spot" to that odor. Interestingly, Mike Rowe found that he couldn't detect certain smells (and others registered strangely) when shooting the odor analysis segment (season 3, ep. 14).
Sadly there are more serious consequences to this than simply being unable to tell if someone's had asparagus. There are poisonous gases which some people can't smell and so can't detect as leaking when they enter a room, which may have debilitating or fatal results.
Certain high-frequency sounds - between about 15 kHz and about 20 kHz - can be heard by children and adolescents, but generally not by anyone older than 25. Sounds in this range have been put to use both as teenager repellent (because by nature it's an extremely irritating high-pitched whine) and as an Inaudible To Adults mobile phone ringtone. As time passes, a person's hearing gets damaged and therefore lose some of their range. Those who take care of their hearing (protect them from loud noises, mostly) and/or have sharp ears to begin with will usually keep their ability to listen to these sounds. Conversely, some can never hear into this range, by a simple quirk of genetics.
Those inaudible mobile phone ringtones have one (occasionally hilarious flaw): the adult in question (generally a teacher) may not be able to hear it, but when everyone in the class is clutching their ears in pain and yelling at the person whose phone is ringing...
Certain bitter chemicals, such as phenylthiocarbamide (PTC) can only be tasted by people with the right flavor-receptors in their taste buds. People with the greatest variety of receptors and/or a denser supply of them are known as "supertasters", and tend to dislike flavors with bitter components, such as coffee or cabbage.
Colorblind people can be useful in aerial recon. Camouflage is largely dependent on matching the colors of the surrounding environment to blend in - so some forms of camouflage don't work on individuals who don't see colors normally.