In The Elder Scrolls V: Skyrim, we actually get to use the Thu'um, where it is explained that it is literally a Language of Magic that makes real whatever is spoken in it. The Thu'um uses the dragon language, as dragons naturally have the power to alter reality by simply speaking. When they're "breathing" fire, they're technically commanding fire to come into existence, and battles between dragons are essentially very loud debates. (Gives a new meaning to Flame War).
It is explained that anyone can learn to use Thu'um, but it takes a great deal of training, mostly to learn the true meaning of the words in the shout. Anyone can try speaking it, but you need to put your soul into it for magic to happen.
We also get to meet the Greybeards. When they call for the Dragonborn, all of Skyrim can hear it. When they whisper, it shakes the mountain they live on.
The Dragonborn is special because s/he can use the Thu'um without any training after absorbing dragon souls, thanks to being the spirit of a dragon in a mortal coil. Technically, the Dragonborn could just casually say Shouts like 'Fus Ro Dah' or 'Lok Vah Koor', but s/he always shouts them, presumably because it sounds cooler.
The Mercenary (Melee Area of Effect) class in Dragon Nest has an entire range of shout skills. One of the higher level ones actually creates a vacuum when he inhales to bring enemies closer so that the actual shout can hit more of them.
Pokémon has a few attacks like this, such as Uproar, Echoed Voice, Noble Roar, Disarming Voice, Snarl, and Hyper Voice. There's even one called Round which becomes stronger if both Pokémon in a team battle do it at the same time. There are also sound-based status-affecting moves such as Screech and Supersonic, and the Pokémon ability Soundproof can protect a Pokémon from any sound-based attacks. Howl can boost the user's own Attack Score, while Roar can scare a wild Pokémon away or force a trainer to switch Pokémon. The evolutionary line of Whismur/Loudred/Exploud and Noibat/Noivern are also all based on this trope, and both can use the most powerful shouting attack, Boomburst.
Queen Sindel from Mortal Kombat. Notable in that she can scream the flesh off people.
Redeads in The Legend of Zelda can stun people with their screams, which get higher and more shrilled over the years.
Skull Kid in Majora's Mask, whose scream speeds up the falling of the moon.
Pol's Voices can be stunned by the player shouting into the microphone—on systems that have one, that is. Phantom Hourglass was the first game where this was possible outside Japan, but it could be done in the original The Legend of Zelda using the microphone built into the Famicom's second controller.
Warriors in World of Warcraft get a few "shout" attacks. They don't damage, but at least one causes "fear".
As of Mists of Pandaria, Warriors can pick a talent called "Dragon Roar", which on top of being a possible Shout-Out to Skyrim, deals fire damage to all nearby enemies.
Warlocks also have Howl of Terror (causes fear), priests have Psychic Scream, but the name implies it to be a magical effect rather than actual noise.
In Guild Wars Shout skills are simply skills that have no cast time (and, indeed, can be used while you're casting or using another skill) and have an effective range of "earshot" (which is approximately aggro range). Their effects are generally damage modification (either increasing or decreasing it), but one shout, "You Move Like A Dwarf", actually damages, knocks down, and cripples whichever foe you have targeted when you use it.
Similarly, The Lord of the Rings Online has many, many damaging shouts. Minstrels alone get at least three ways to kill via yelling, with bonus points to Dwarven minstrels, whose shouts, rather than wordless "AAGH!" of every other race, consist of the war-cry "Baruk Khazâd! Khazâd ai-mênu!"
The barbarian of Diablo II has a number of vocal capabilities (howl, taunt, shout, battle cry, battle orders, war cry and battle command) with a number of effects on enemies ranging from fear, stunning or status penalties all the way to immediate damage. And they can grant allies temporary buffs.
In Condemned 2: Bloodshot, there's a whole cult with this ability. You're one of them, and gain it for the last level.
Mega Man and Shade Man had the sonic weapon "Noise Crush" in 7, but that came from their Arm Cannon.
One variation of the Trumpy Virus produces a blast of music that, while it does not impair you in any way, will still prove an annoyance, by completely refilling a fellow mook's (or it's own) HP. As it is usually teamed up with viruses with sizable HP, as well has having a lot itself, you WILL want to defeat it first.
In a twist on this trope, the Street Jam psych from The World Ends with You is triggered when the player stimulates the DS' mic. It comprises of Neku summoning giant speakers that blast sound into every Noise in the current battle. A variant, the Twister psych, turns mic stimulation into gusts of wind that blow enemies up and then slam them down.
In Command & Conquer: Red Alert 3 the Allied attack dog and Soviet war bear have functionally-identical stun abilities based on their roar/bark. The dog has an amplifier to help, and can pass this along to the Multigunner IFV if the dog hops in the shotgun seat. The Allied dolphin (also in Red Alert 2) has an amplified sonic weapon as an actual damaging attack that can take out boats if they attack in groups.
In Tiberian Sun, the GDI had the Disruptor tank which was derived from Scrin technology. It fired high powered sound waves at targets, and was one of the more powerful units in the game. In Tiberium Wars, GDI used sonic weapons as static base defenses, and later built smaller mobile versions as well. According to the backstory, this technology was useful in combating the spread of tiberium, though it doesn't seem to have much of an in-game effect.
Universe at War has the Amplifier for the Novus faction, which basically functions as a sonic artillery unit.
Knights of the Old Republic II: The Sith Lords has the Force Scream power, which is one of the best Force powers in the entire game. note At master level it deals 7d6 points of damage AND reduces all attributes by 6 points (-3 modifier), thus reducing the enemy's Vitality, Force points, damage, defense, chance to hit, saves...pretty much everything. Not to mention it is tougher to save against (10+level+Wisdom mod+Charisma mod rather than 5) and deals area damage.
The sirens in God of War II have a migraine inducing attack of this nature.
BioShock Infinite, meanwhile, has two monstrous enemies who have screaming as a big part of their motif. The Siren's voice is so powerful she can tear through the veil between worlds, allowing her to summon the souls of the dead — she uses this to resurrect the enemies you kill. The Boy of Silence, on the other hand, is a living alarm system whose freakish scream draws swarms of enemies to kill you.
In New Super Mario Bros. Wii, Bowser does this in the final battle, but it doesn't do any damage, just pushes you back a little.
Bowser Jr. uses sound waves as part of his super ability and mega strike in Mario Strikers Charged.
Rikuo/Aulbath from Darkstalkers has a special move called Sonic Wave, where he emits ultra-sonic waves to stun his opponents and that inflicts some damage on them.
City of Heroes offers Sonic Attack and Sonic Resonance as powersets. The default animation for these powers are a range of screams and shouts. Oddly, while Sonic Attack damages targets and renders them vulnerable to more damage, Sonic Resonance includes a number of powers that make your allies tougher instead. All by making strange shrieking noises.
One of the warrior's abilities in Dragon Age: Origins is "War Cry". When upgraded it can knock enemies off their feet.
Some minor enemies in Golden Sun games can use a vocal attack. Bats unleash sonic waves, Harpies let out a shriek, and Scuttlersbellow a bloodcurdling cry. These deal damage and sometimes prevent an action or even stun a character. Also, until Dark Dawn, unleashed weapon techniques were narrated as "(character)'s (weapon) lets out a howl! (technique name)!"
Video Game/Warframe has the Banshee warframe, which primarily relies on sonic strikes and echolocation to find and kill enemies.
In Brütal Legend, General Llyonwhite does this to a mirror falling on him. Unfortunately for him, his scream breaks the mirror and the huge, sharp shards kill him
In Monster Hunter, the majority of the monster's roars have the ability to stun you and, for the most part, can only be avoided by rolling or blocking with your weapon. However, some monsters (such as the Tigrex) have roars powerful enough to inflict minor damage and send you hurtling dozens of feet across the landscape.
In Archon, Banshees have this as their power. It has very short range, but they are the only melee unit that can move and attack at the same time.
In Dune II, after getting the IX building upgrade, House Atreides can manufacture Sonic Tanks, capable of hitting multiple targets if they're all in the line of fire.
In the flash-based webgame Swords and Sandals 2, a valid min-maxing strategy is to focus mostly on Charisma, so that your Taunt skill becomes your free, deadly, ranged, and only, weapon.
Dead Rising 2: Metal band Angel Lust obliterates an entire horde of zombies by blowing up their heads with a deafening Big Rock Ending. This specific trope would best apply to Jeanna, the band’s sultry siren of a lead singer, but her Metal Scream is only implied, and never actually heard on the soundtrack. Rescuing the band allows the player to craft a "Power Guitar", a combination of an electric guitar and amp that will blow the heads off of zombies with The Power of Rock.
Orc barbarians in Heroes of Might and Magic V are forbidden from learning magic, instead they get a selection of battlecries that function as buffs for their soldiers.
In the Spin-OffPersona 4: Arena, Kanji Tatsumi's standing heavy normal attack is him screaming "You want some?" with a large speech bubble appearing behind him and damaging the opponent if they touch it.
Kyouko Kasodani from Touhou is able to manipulate sound waves. Her attacks are normally centered on her amplifying her own voice to ungodly levels and stunning her prey with infinitely-repeating echoes. On a related note, the Prismriver sisters' attacks include actual visible sound waves in the form of musical notes, but their powers are limited to what their instruments can do.
In After The War, certain hybrids will attack you with a cone of loud sound, which will also knock down even enemies caught in the middle.
Atlantis No Nazo has a microphone powerup that lets the player stun all enemies on the screen by shouting into the second Famicom controller.
Lakelurks are based on the Mirelurk King from Fallout 3. He has the same attack, and it does hurt a lot (it ignores armor).
New Vegas also has the Sonic Emitter in the Old World Blues DLC, basically a gun which fires sonic blasts that play hell on anything mechanical, and can disable forcefields. You can also bring more samples to one of your AI helpers so he can tinker with your emitter and change the blast's effects. Or, in his words, "Get me a good sample base to work from and I can whip up a wave that makes Jericho look like a kazoo".
If you ever so sadly happen to pull a mandrake out of pot without the earplugs equipped in Mad Father, it unleashes a hellish scream that can kill Aya.
Angry Birds 2 gives Red the ability to give whatever's in front of him a shove with his battle cry, which can help topple structures.
The titular Charlie Murder and one of his fellow bandmates Kelly scream into a microphone to use magic attacks, which can be upgraded over time.
The House of the Dead: OVERKILL has the Screamer boss who, well, screams at you to attack. Apparently, this has the side-effect of messing with G and Washington's minds as they begin to hallucinate during the battle.