Here we have the ghostly women who originate from Celtic Mythology. They are usually noted for their voices, due to either being hauntingly hypnotic, or filled with horrible screams usually Sadly Mythtaken for a sonic attack. Traditionally, hearing a banshee wail is a Portent of Doom.
They are also prominent in Irish legends, where many of the more well-known banshee stories come from, as being a type of fairy typically connected to a single family. Their screams were due to mourning for members of the family who died, which often served to inform others of their loved one's death. There have been stories of banshees haunting families even as late as the 19th Century.
In modern fiction, along with the above-mentioned sonic attack, they are usually depicted as a type of undead or ghost, in contrast to their original origins as a type of The Fair Folk. A clue to their origin can be found in their name; bean sídhe can be translated as "fairy woman".
Marvel Comics' Banshee was a male mutant with yelling and bizarre flight powers. His daughter, incidentally, was originally called Siryn. Having accepted her father's death, Siryn has taken on his name — including a Lampshade Hanging that she was never quite sure why he named himself after a female spirit in the first place. X-Men writer Roy Thomas actually meant for Banshee to be a woman, but Stan Lee decided that a male character would be better. When Siryn was created, Thomas' reaction was "that was what Banshee was supposed to look like all along!" So, with Siryn's taking on the name, the Banshee he'd created has finally arrived... and it only took thirty years. But hey, Legacy Characters can take a while.
Corto Maltese once meets an Irish girl named Banshee, and asks her why she was given such an inauspicious name. She dodges the question, but she isn't, as far as the story goes, a supernatural creature.
The DCU has Jeanette, of the Secret Six, and Superman foe Silver Banshee. The banshee of the DCU are typically mystical in origin, receiving immortality, superstrength, and a hideous scream that can kill those who hear it.
Of course, Silver Banshee has to know your True Name. Her power doesn't work on Superman because she doesn't know his birth name is neither Superman nor Clark Kent, but Kal-El.
Depends on how you define "doesn't work". It won't kill him, but it will cause some of the most extreme pain you can imagine. An absurdly loud sound, at close ranges, being heard by someone whose ears are sensitive enough to pick up a whisper from across the city? Not fun. However, if she does know your true name, her scream is less "standard sonic attack" and more instantaneous death.
In one Dylan Dog story he meet a girl named Banshee, who brings death and bad luck to all those who are close to her. Of course, our hero tries to seduce her and break the nefarious curse by surviving himself.
Disney's Darby O'Gill And The Little People had a banshee that appeared and wailed to warn that someone was about to die. It also summoned the Cóiste-bodhar (Death Coach) to take away the soul to the afterlife. It's generally considered a standard-bearer of childhood terror.
In Harry Potter, a Boggart turns into one when it's Seamus' turn; she's described as 'a woman with floor-length black hair and a skeletal, green-tinged face'. Seamus uses the Riddikulus spell to make her lose her voice.
Discworld gives us two different varieties of Banshee, the only sentient species on the Disc that evolved natural flight. There's the civilized type, which typically wails when someone is about to die (the one we meet has a speech impediment, so he just slips a note under their door). Then there's the feral variety, which are the reason for the legend—they also wail when someone's about to die, but in this case it's generally because they're cutting out the middleman and hunting you down themselves. The two we've seen (one of each) are both male.
Mercedes Lackey's SERRAted Edge novels have a male Banshee as an enemy of the elves, with a painful false etymology (mixing Gaelic with Old English) that banshees are the bane of the Sidhe.
In The Icewind Dale Trilogy book, The Halfling's Gem, Drizzt do'Urden and Wulfgar have to fight a classic D&D-style banshee. Both sides survive, because the local village actually relies on the fact that a banshee lairs nearby for tourism, and Drizzt and Wulfgar promised not to destroy her. This is a plot point because her lair holds the enchanted mask that lets Drizzt pass as a plain old elf to avoid the obvious problems of being recognized as a dark elf on the surface and that eventually falls into the hands of Artemis Entreri.
Soul Screamers has Kaylee who carries on the banshee wail whenever someone is near-death.
Erica Hayes's Shadowfae series of urban fantasy novels features banshees with siren-like abilities. These banshees seem to be a type of fae with a magical affinity for sound rather than death, although they can kill with a song or scream if they want to. They can also cast a variety of spells through song, manipulate humans, and secrete venom from beneath their tongues that they can use as an additional weapon besides their voices. Their magical affinity for sound gives them preternatural hearing but also gives them the ability to filter sounds so they don't get overwhelmed by auditory stimuli, as well as enhancements to their inner ear that give them extraordinary balance and agility. They all look like attractive human women, but with unusually colored hair and eyes, and usually their eyes and hair are of different but unnaturally bright neon or metallic colors. They are known for being violent, lustful, and usually just a little psychotic, and in one book, a banshee is employed as an enforcer and bodyguard for a supernatural version of the mob. They are born with innately magical voices, but their magic can be taken from them by some other kinds of supernatural beings who use the banshee's voice magic for themselves.
In Charmed banshees are spirits attracted to heartbroken humans; they use their high-pitched screams to kill them. If they use their screams on a witch the witch turns into a new banshee.
In Lost Girl, banshees are a type of Fae who get involuntary premonitions of death. They don't consciously know who, how, or when, only that it's someone around them, that it has to be a member of one of the ten Noble Families (five human, five Fae) and that it will be soon. They do keep the details subconsciously, however, and it can be forced out of them using iron, to which they're highly allergic. In the relevant episode they used a liver shake.
In the Disney show So Weird Fi fears that a banshee has come for her Irish grandfather.
In Teen Wolf, Lydia is eventually revealed to be one. This comes with the ability to sense death and a piercing scream although, thus far at least, the scream is more of an extra effective "help someone's trying to kill me!" alert than an attack.
In Warhammer 40,000 Eldar have a unit of berserker called Banshees that self-identify as female regardless of biology, armed with high frequency emitters that allow them to stun their prey with screams.
Banshees show up in Dungeons & Dragons. They're pretty nasty. There's even a spell called wail of the banshee, which is basically a "everyone within this radius of the caster, save or die" spell.
Warhammer has screaming undead banshee as a unit in Vampires army.
The original Celtic breed show up in Scion Companion, under their original name, bean sidhe ("sidhe" is pronounced "shee"). Since White Wolf Did The Research, they're fixated on death but aren't particularly big on screaming.
Banshees are a Shade of ghost/projector in Orpheus. In keeping with the name, their major talents are the ability to see the future and a wail that can either control emotions or shatter your eardrums.
Warcraft banshees are Undead units who attack with their high-pitched screeches. They function as spellcasters, making enemies miss, rendering units invulnerable to magic, or possessing enemies. They were once High Elves whose bodies and souls were defiled by the Scourge, forcing them to exist as bitter, spiteful ghosts.
The most notable Banshee is Sylvanas Windrunner. Originally forced into a ghostly state as a final cruelty by Arthas, she was the first Banshee and became the "Banshee Queen". As a reward for her service to Arthas, she eventually received her original(now undead) body to possess.
They also show up in World of Warcraft, unsurprisingly. Part of them keep their long-range wail attacks while others melee the player, most still use curses that reduce stats or make the target miss. At least one (a boss) can temporarily possess players.
One of the enemies in the English version of Castlevania: Order of Ecclesia is called the Banshee (in Japan, it was an onryo). One quest requires you to record its scream on a phonograph.
The "Frozen Lady" of Mystery Case Files: Dire Grove is also identified as a banshee. She doesn't have a scream attack, but when you can freeze a good chunk of England solid while still mystically bound, do you need one?
Misdreavus and its evolution Mismagius from Pokémon are a relatively lighthearted take on banshee, as they use their cries to scare others for fun as often as they use them to battle. Perish Song, which causes both combatants to faint if they listen to it for three rounds, appears to be their Signature Move, though other Pokemon can use it. They're portrayed as ghosts/witches rather than fairies, and can be male.
Mortal Kombat's Sindel fits the criteria thanks to her Banshee Scream attack and ghostly appearance. She even returned from beyond the grave in her debut.
In Mass Effect 3, the Reapers turn humanoid species into cyborg-zombies. When it's done to the Always Female race of Blue Skinned Space Babes with Psychic Powers, the result is called a Banshee for good reasons. In addition to a psionic scream that targets any nearby creatures, they can also cross large distances in a fraction of a second and make turns around corners, making it very hard to stay away from them.
Joker: Mutating people to turn them into living weapons is one thing, but the yelling? Why make them yell?! That's totally uncalled for!
Blood has ghosts which screamed. get used to AAAAAH! AAAAH! AAAAH! as you had to fight these things. Their screaming didn't do any damage but holy hell, it was loud.
Gargoyles features a villainous banshee who is one of Oberon's Children. She has a small cameo in a second episode when she refuses to answer Oberon's summons; as punishment she's dragged back by the Weird Sisters and gagged indefinitely.
The Extreme Ghostbusters fought Banshee sisters; one was the stereotype, the other could use her voice to hypnotize people.
It was a trap: The Siren was the carrot, the Banshee the stick. In the episode the banshee & siren could stay young if they stole the youth of humans. So they would use the siren to lure young people in and the banshee would then steal the youth.
In Scooby-Doo! Abracadabra-Doo, Scooby and the gang face a number of threats, including a banshee that was imported to America because the castle it had been associated with in Ireland had been brought, brick by brick, to the U.S. Shockingly, in the Scooby-Doo tradition, it turned out to be a fake.
In The Real Ghostbusters episode "Banshee Bake a Cherry Pie?", the group had to stop an Irish rock star who actually was a banshee and was scheming to use her powers to destroy the entire United States through a broadcast concert.