Former Teen Titans member Herald gained this power through a super voice box. Unfortunately, as Vox, he can't turn it off — while he can control sound, anything he actually says will kick the ass of everyone in the area. Not as bad as Black Bolt, though — see below.
Spider-Man has a couple of characters with this set up.
Most obviously, there's 90's villain Shriek.
Phil Urich, the heroic Green Goblin was indicated to have a "Lunatic Laugh" that leaves people stunned in pain and is powerful enough to harm symbiotes, which are weak to sonics. Interestingly, the power was premised on the idea that the Green Goblin (whoever he/she was) had a creepy high-pitched voice, but more recently, the Green Goblin is presented as Evil Sounds Deep.
Now that Phil is the current Hobgoblin he's back as a cackling, shrieking villain.
B-lister villain the Shocker, despite sounding like he uses electrical attacks, actually uses a harness to create vibrated waves of air that literally shake things apart, leaving him somewhere between here and Blow You Away.
The X-Men's Sean Cassidy aka Banshee and his daughter Theresa aka Siryn. Screaming also lets them fly, somehow. Oh, and they can talk while using it. Hilarity ensued in the animated series when Banshee had to give Wolverine a ride.
Wolverine: With you making all that noise, we might as well try sneaking into this place with a marching band.
Banshee: If you think that's bad, you should hear me sing.
The all-time champion is probably Marvel's Black Bolt who can level buildings just by whispering because his voice isn't just super-sonic but actually causes massive disruptions in electrons. When he declares war, he's not mixing metaphors. In an Alternate Universe, he committed genocide by yelling at the top of his lungs. However, since he cannot actually control it — as in any whisper from him will destroy buildings, etc. — this makes his position as King of Attilan and leader of the Inhumans extremely reliant on his cousin and wife Medusa's ability to read his body language. He's gotten so good at laser-guided body language that characters have commented on how uncanny it is that they always know what he means.
What If? #34 (the Rule of Funny issue) asked: "What if Black Bolt got the hiccups?" [cue destroyed Manhattan]
Cosmo, the psychic dog from Nova, seems to have a sonic bark.
Marvel's Iron Fist ranks the "Silent Scream" among his "upper-tier abilities". A hypersonic attack that can KO an entire room of thugs, it also leaves him unable to speak for a month.
Songbird of the Thunderbolts in the Marvel Universe was formerly Screaming Mimi, who had traditional sonic powers. After her lover, Angar the Screamer (whose screams could induce hallucinations) died, she screamed so loud that she essentially melted a forest and blew her throat out. Zemo recruited her and gave her a harness that turned her screams into energy constructs, using technology pirated from the supervillain Klaw, who does the same. Of course, Klaw was not thrilled to learn of this theft... Later on, in the New Thunderbolts title, Songbird temporarily lost her powers while using them to hold up the UN building, but eventually got her voice back.
Did we mention Klaw? Ulysses Klaw can use sound as a battering ram, but he prefers to turn it into energy constructs, making sound a Green Lantern Ring. Turns out that his own body is a sound construct: the same sort of tricks or Applied Phlebotinum that will disrupt his powers cause him to vanish, leaving behind only the loudspeaker-looking device he has for a hand.
The Wolverine story "Roar" features a beast whose roar is so powerful it leaves an entire town with bleeding ears, and when Wolverine shows up he has to keep regrowing his eardrums to fight it.
The Authority arc "Godhead" brings us Sister Voice, who leveled Salt Lake City with a powerful rendition of Paul Anka's Puppy Love.
Howard the Duck's enemy Doctor Bong uses sonic weapons, but he doesn't shout. Rather, his helmet is shaped like a bell, while his severed right hand is replaced by a giant clapper; by striking the bell-helmet with the clapper, he produces dangerous sonic projectiles.
The Swedish comic James Hund once featured the Tibetan Scream Ninjas, who practiced the "especially Oriental" art of screaming buildings apart.
Accomplished Perfect Physician of the Great Ten has a variation on this; the original Perfect Physician devised a way to use sound to find and heal ailments, using a mere click of his tongue to cure cancers. His successors extrapolated this ability: the second learned to stop earthquakes with a growl, and the third turned back a Mongol horde with a shout. Because the collective memories of the Perfect Physicians are passed on to each new title-holder, successive Perfect Physicians have been able to perform all these feats and more.
In a Darkwing Duck miniseries, one-time character Screaming Mimi could send several men flying backwards with one scream. Alas, she wasn't so useful once they learned to squirt water into her mouth.
Tyroc of the Legion of Super-Heroes has a variety of effects that he can make with a variety of screams (there's a different sound effect for each... effect.)
Long time Superman foe Silver Banshee is not only loud, but if she knows your real name, then hearing her scream results in instant death. She can't kill Supes because she doesn't know his real name, Kal-El. She can still blow out his eardrums, though.
Fellow banshee Jeanette, of the Secret Six, can leave you comatose with her screams.
Superman, usually only displayed when he loses someone close to him.
In Final Crisis, Superman finally kills Darkseid by screaming so loud that the evil god shatters. The effort nearly ruins Supes' vocal cords.
The New 52 version of Supergirl displays this on occasion, first when attacked by the Powered Armour wearing mooks of Simon Tycho, then when fighting Black Banshee. He specifically notes it and mocks it as a pale reflection of his own powers, before giving her a demonstration.
In Buzz!, this is the power of the Spelluminati champion. The sound of their screams is enough to temporarily deafen the Black Queen. It doesn't affect Webster though, because the champion is also his sister and thus he's used to her screaming at him.