In Haibane Renmei, the Haibane have small wings. They can move a little, but never fold or really flap. The wings are always held stiffly behind them or to their sides.
The series' trailer has an extremely quick flash of an X-ray of a Haibane wing, which indicates that only half of the wing bones are outside their body; the other half, required to fold them, lays flat in their backs. A fan theory suggests that they break out all the way when the Day of Flight arrives.
There are several Winged Humanoids in Tower Of God who can't fly, leading to rather funny situations when Androssi knocks people off platforms.
Apparently those wings were not made for flying…
In one episode of FLCL Khanti is seen wearing little black wings and an obviously fake halo. They don't do anything except convince Mamimi that he's a god...
In the One Piece Skypiea arc, the Shandians and Skypieans all have funky wings on their backs that do nothing.
Subaru in the .hack anime has tiny wings. Her character is a Digital Avatar, so it kinda makes sense.
When Signum of Magical Girl Lyrical Nanoha Striker S gains a new Unison form at the end of the show, she gets a spiffy set of flaming wings to go with it. Of course, since she could already fly even before that, they're mostly just there to look cool.
Hayate's black wings also seem to only be there for the Perpetual Molt effect, though the Book of Darkness, the original owner of those wings, did use them to help her with her initial lift-off.
Wings that do something are viewed as a "power" in Saint Beast, meaning most angelic characters have wings that do nothing if it's not granted to them. These wings tend to be hidden as such.
Subverted in Angel Beats!!: Tachibana uses her "Angel Program" to create wings to make her seem more "angel-like." They're beautiful and seem to be mere decoration—until the moment she actually uses them to glide off a building.
There's a half-example in Rozen Maiden where Suigintou's wings do something—act as weapons—but have absolutely nothing to do with flying.
Cell's wings in Dragon Ball Z are a real waste in a series where flight is all about Ki. He flexes them occasionally, but the fact that he loses them in one form says everything.
In Mai-Otome 0~S.ifr~, Lena Sayers has extremely long wings as part of her psudo-HiME powers. It's not clear if they have a purpose or not since she can fly anyway, and the only time we see them is right after her robe breaks.
Gundam doesn't seem to be sure whether it wants to play this trope straight or subvert it. In some cases (such as Destiny, Freedom, Deathscythe, Aile Strike, Force Impulse and Zeta Gundam), the wings are just there for show, though Freedom does hide two guns in them, and Strike Freedom uses them to store its DRAGOON system. In other cases (such as the Tallgeese, Wing and Wing Zero) the wings actually serve a purpose, as they contain several thrusters inside to increase speed. They're still static, except for Wing Zero in Endless Waltz.
This is the case for the Angel and Imp races of humans in Centaur no Nayami. Since they can't use their wings to fly, there are in-universe theories for why else they would have evolved - protection from harsh weather is one.
After Medusa's departure from the Frightful Four (obvious Evil Counterpart to the Fantastic Four), the Wizard sought a new supervillain to replace her. Among the applicants was a fellow named the Osprey, who had nifty-looking wings attached to his back. When the Wizard discovered the wings were nothing but decoration, and that the Osprey had no powers at all, he stuck an anti-gravity disk on the fellow and sent him floating helplessly into the upper atmosphere.
The hippogriff in The Prisoner of Askaban has wings that flap, but based on how smoothly he rises, they do not cause him to fly.
The Lord of the Rings: It's ambiguous as to whether Balrogs actually have wings. If so, they didn't save the Balrog from falling down the pit with Gandalf.
In one of Bruce Coville's Book of Magic anthologies, the story provided by Coville is about a teenager who wanders into Elives' Magic Shop and ends up purchasing a home magic kit thematically inspired by the metamorphosis trick of stage magic shows. Rather than switching places with an assistant, he finds that it gives him wings-which don't work. He continues to follow the instructions, but grows increasingly agitated as it's becoming harder and harder to hide them from his abusive uncle, and they still don't let him fly, so his dreams of getting away from said uncle are as kaput as ever. Fortunately for our protagonist, he manages to keep them under wraps until he's finished the process, after which they fill out and do let him fly, and by the time his uncle finds out, he's already headed out the window.
In the book Storm Thief there is a character named Vago, who is a giant genetically engineered golem with wings, though technically they do let him glide he can't actually fly. This is lampshaded/justified by The Dragon when he says that he had overseen Vago's creation and insisted that they give him wings despite the technicians telling him that Vago was too heavy to fly but was able to glide on strong wind currents.
Peregrine in the Wild Cards series while often called an Ace is actually a Joker-Ace, i.e. someone who was both physically mutated by the disease and gained superpowers as a result. Her wings would allow her limited flight, or at least gliding, but she actually uses specialized telekinesis to fly. She avoids the stigma of Jokerhood because her wings are pretty, unlike most Joker mutations.
In Magic: The Gathering, Whippoorwill is a bird and has wings, but cannot fly. Art guidelines have been changed so as to prevent this kind of confusion in current sets.
Subverted in an older card, the Winged Golem's flavor text tells that the wings were meant to be ornamental, and it learned to use them by itself. For a small mana cost, it can fly for one turn.
In Changeling: The Dreaming, you can have wings as either a merit or a flaw. As a merit they do indeed allow you to fly, while as a flaw they get in the way a lot and make clothes shopping hard (but they look cool).
In Wraith: The Oblivion, some wraiths will spontaneously develop vestigial wings after waking up dead. This, and other physical alterations, is related to their psychology, somehow.
In Witchcraft's follow-up game Armageddon, angels and fallen angels can have wings or not and are able to fly either way. However, having wings does improve maneuverability in the air.
Fable III will feature wings that pop out of the Hero's back, and so far as anyone can tell they exist only to look awesome and show just how badass good/evil the Hero is.
Also, some versions of Moogles can fly, but others can't (including almost all of the ones in Ivalice) yet have wings anyway.
The moogle in Final Fantasy XIII-2 can fly, but it's stated that it flies by magic, and that its wings are just for show.
In Castlevania: Dawn of Sorrow, a Winged Humanoid soul gives Soma Cruz visible wings. They increase his luck. That's it.
Averted in Order of Ecclesia, where Volaticus does the same thing - but Shanoa can use them to fly. Good luck doing Large Cavern without them.
The Reapers and Angels in The World Ends With You. Both have wings and can fly, but use not their wings to fly, but their Psychic Powers.
Tabuu in in Super Smash Bros Brawl gains wings for precisely one attack. They don't seem to do anything (given that he seems to be able to levitate and warp for short distances, anyway), but since Word Of God says that Sonic weakened said move by attacking them, they may serve some purpose. Since there's very little reason for them to be wings specifically, though, they still count.
Before the wings were weakened, they could take out everyone(!) with one attack, so while the wings aren't for flying, they serve a purpose. The attack was basically at will too, so only a sucker-punch from someone really fast who hadn't appeared up to that point could do much about it.
Many games with Virtual Paper Doll features (e.g. Guitar Hero 5 and Tony Hawk's Underground 2) have wings as character creation items. Even Mortal Kombat Armageddon, which had characters that could fly, gave your character the possibility to add wings to their design, but not a move to put them to work (not even a mo-capping from Sindel, who can fly without wings).
Everyone in the world of Baten Kaitos has wings, which allow them all to fly. In battle. Outside of battle, they still use stairs, ladders, etc, and most buildings are one or two stories. This is subverted in Baten Kaitos Origins, where characters fly around with wings outside of battle, and the main character uses them to dash about and jump over large gaps.
And yet, they still are reluctant to build anything with more than two floors, or put doors anywhere but at ground level.
Early in the first game, you're told that the wings aren't as functional as they look - short flights, like in battle, are fine, but you're going to fall a long way if you try to get any further than that. The wings used to be more functional, but over time they've atrophied. That's why they have flying boats. As for buildings, it would be an unnecessary mindjob to draw up building plans with the door on a higher level (you have to take into account that there's going to be a very large hole on an upper wall). Just because they have limited flight doesn't mean their entire society is based around their wings.
There's also Demon Hunters in their demon forms, who turn into typical looking demons (except they're made of black fire), complete with bat wings, which of course do nothing. Most ridiculous case are the Pit Lords, with their centaur like set up, have wings on the humanoid part of their bodies.
Dragon Disciples in Neverwinter Nights gain wings at level 9. Flight is not implemented in the game, so these wings do nothing.
Kaelyn in Neverwinter Nights 2 Mask of the Betrayer and Persey in the Tales of Arterra mod both explain when asked that their wings aren't powerful enough to fly with, but make wonderful waterproof cloaks when it rains.
In Kid Icarus, the hero starts out with only decorative wings, but acquires flight power in the final stage. In the Game Boy sequel, they can slow his descent and Pit is again able to obtain a powerup that allows flight. In Uprising, he's still incapable of flight, and Palutena has to personallybestow the "Miracle of Flight" upon him, in order to let him fly for 5 minutes at a time.
Several characters in Baroque have wings, but are never seen flying—not even the Archangel, who has the only wings that look big enough to be functional. The protagonist can also acquire (fake) wings which grant various effects (immunity to Poison, Vitality regeneration, etc.), but never the power of flight.
In LEGO Star Wars, you can play as Watto, the Toydarian junk dealer from The Phantom Menace. His normal mode of travel is flying, but don't try it over a chasm...
In the various Atari Godzilla games (like Godzilla Destroy All Monsters Melee), the creators made a point to ensure that the only flying monsters were those who rarely touched the ground, like Rodan, Mothra and Megaguirus. While this isn't a big deal for burrowing Megalon (whose wings only appear when he opens his shell) or teleporting Gigan (whose ability to fly in the films was due to jet boosters not seen on his design) Destoroyah sports not one but two pairs of enormous, bat-like wings that do absolutely nothing. This is especially frustrating, since the movie Destoroyah spent half its onscreen appearance in the air.
In Star Ocean: The Last Hope, your winged White Mage Sarah has huge wings but is described both by the game and in her own words as being incapable of flight, though she can glide. In any case her ending has her get over this and properly fly.
Dwarf Fortress has a fairly unusual example with cave dragons: they have wings incapable of flight, which their description implies worked before they became adapted to the underground. But the thing is, regular dragons don't have wings at all, which implies they're either descendant of a type of dragon not in the game or that they grew wings which served no purpose after coming underground.
Toady has said that dragons will have wings to fly at some future point. He's waiting until he can program them to use it to devastating effect.
Some titans and forgotten beasts don't fly even when they do have wings, while demons do whether or not they have them.
In Etrian Odyssey III, the Zodiac class features large, artificial, crystalline wings in all of its character portraits. While they appear to allow the Zodiacs to hover in place in said portraits, they don't serve any kind of purpose whatsoever in gameplay.
The Pokémon Druddigon. Despite having wings, it's not a Dragon/Flying-type—it's just a Dragon-type and doesn't have Levitate. According to the Pokedex, it uses its wings to take in sunlight for warmth.
The ostrich-like Doduo and Dodrio are inversions. Despite having no wings, they can learn the move Fly.
The original Japanese name for Fly can be translated as both "Fly" and "Sky Jump".
In SSX On Tour you can get bat wings as an accessory. This Troper has had to explain to suspicious multiplayer opponents many times that, in fact, the Wings Do Nothing.
Bug! has the titular character. He's a bug that has wings, but they're too small and thus he is unable to fly. In the sequel Bug Too!, he gains the ability to hover for a short distance, but he hilariously flaps his arms to do so instead of using his wings.
Slightly Damned has Sun Pendants, which grant angels their wings. However, when Buwaro, a demon, ends up owning a sun pendant with no anti-demonic defences, it gives him stylish but obviously non-functional miniature wings.
Alien Dice: Lexx has wings, but never flew outside battles.
Subnormality: The Sypix has wings, but walks everywhere. Justified in this case as she have to file flight plan each time, and she didn't want to file each time she went somewhere.
A lot of on-line avatars and self-portraits do this.
With Strings Attached: the John Lennon of New Zork (as opposed to protagonist!John) has wings but is otherwise human, so he can't fly.
The guardians in W.I.T.C.H. all have small wings in their magic forms. While they do flap when they fly, they are actually useless, as evidenced by the episode where they lost their powers and instantly dropped out of the sky.
In the comic Hay Lin was the only one who could even fly at all.
Subverted in The Venture Brothers - eventually. In the first season the Monarch and his henchmen's butterfly wings seem to be purely decorative - the writers intended for the Monarch using his wings to fly in the early season 2 episode Hate Floats to be a big deal, but this didn't come across in the animation - then in the season finale, the henchmen discover, to their shock, that they can, in fact, fly.
"Why did nobody tell us this?! This changes everything!"
Dragon Maleficent from Sleeping Beauty has wings coming out of her back, but for some reason her dragon form cannot fly.
The Twins' father in Superjail! appears as a large, winged balloon-like creature. But due to the tiny size of his wings, they're unable to help him fly when he's injured and generally just flap uselessly as he speaks.
Many flightless birds have highly vestigial wings that are useless for flying. Some species use their wings for other purposes, including penguins, which use their wings while swimming, rheas and ostriches, which use them as running aids and display structures, the kakapo, which uses them for gliding, and the kagu, which uses them for display and gliding, possibly among others.
Deinonychosaur dinosaurs (the "raptors"); many species could glide and even fly, and even species that are considered to be flightless, such as Velociraptor itself, have quill knobs, something that doesn't exist in flightless birds, suggesting that they preserved wing feathers, and likely used them for WAIR (Wing Assisted Incline Running; on other words, flapping the wings so hard that they can run vertically).
An alternate theory is that they used their wing feathers the same ways some modern raptors do - flapping for stability as they clung to prey with hooked claws and ate it alive.
Reproductive-caste ants and termites use wings to fly in search of a mate, but after mating these have no purpose and are shed or bitten off. Most queen ants dissolve the wing muscles for nutrition after shedding. Species where queens are wingless usually can't found a colony by themselves as they're missing the wing muscles, and would starve on their own.
Peruvian-born painter Boris Vallejo has stated himself that the wings he paints on his models are purely aesthetic.
Mesopotamian art often shows gods and other supernatural beings with wings, but they never show them in flight.
Polish hussars were often depicted in art with large ornate wings as part of the uniform. It's unclear whether hussars actually went into battle with the wings and what use they could possibly have. Some theories suggested that they might defect blows from certain angles or that the vibrating feathers make an eerie noise during a charge. On the other hand, so much drag and extra weight would probably counter whatever small advantage they might have provided, making it unlikely that they were anything more than parade dress.
Winged cats: as stated in The Other Wiki, the origin of such "wings" is either matted fur for longhaired cats, a skin disease or conjoinment/birth defect.