Dumbstruck

Sometimes your Heroic BSOD is so bad your language software crashes. In non-geek terms, this means the character that had the Heroic BSOD is unable to talk. Usually resolved by means of Epiphany Therapy.

Anime and Manga

Rose in the 2003 anime adaptation of Fullmetal Alchemist.
Misato from Neon Genesis Evangelion, after surviving the Second Impact and seeing her dad getting killed by an Angel, spent many months like this.
Kisa from Fruits Basket goes mute from being bullied at school and ignored by her only friend (and love interest).
Koji Nanjou in Zetsuai 1989 goes mute after he's in a nearly fatal motorcycle accident. He's in a coma for a while and does come out of it, but when he wakes up he can't talk. This is a problem because he's a singer by profession.
Death Note: One of the effects of Mello's kidnapping of Light's sister Sayu is leaving her unable to talk. And walk. And do pretty much anything, really. By the end, she's starting to get better.
Casca from Berserk suffers the loss of her speech as well as her memory as part of Going Mad From The Revelation after she is raped by Femto during the Eclipse.
Played for humor in One Piece; when the Strawhat Crew's bounties are updated after Enies Lobby, Sanji is so upset with the picture on his (they couldn't get a photo, so it ends up as a Gonk version of him) that he stops speaking intelligibly for a while.
In Full Metal Panic!, Sousuke was shown to have temporarily become mute when he was 3–4 years old, during his backstory. This being after he watched his mother (who died protecting him, telling him that he must survive no matter what, and "fight") fall to her death. He does end up being able to talk after he sees Kalinin (who had been constantly visiting him, trying to cheer him up and coax him to talk) slip and fall. Sousuke finally asks him if he's okay, and Kalinin is overjoyed.
In the very first episode of Fist of the North Star, Lin was like this due to seeing her parents get murdered right in front of her by bandits (most likely Zeed's men). Kenshiro cures her of this using one of the healing techniques of the Hokuto Shinken.
Chapter 10 of Magical Record Lyrical Nanoha Force reveals that Lily Strosek used to be able to speak, but the emotional damage from her life as a lab experiment eventually made her unable to project her voice.
Young Clare in the Claymore Flashback Arc.
Phoebe and Rita Rossi from Ashita no Nadja.

Films -- Live-Action

The character known as Grace in Dead Again starts the movie only able to talk in her sleep, but not at all while awake. It turns out she's having Past Life problems.
Halloween. Dr. Loomis said that after stabbing his sister to death and being institutionalized, Michael Myers "hasn't spoken a word in 15 years."
Scrooged. The Tiny Tim stand-in is a little boy who hadn't spoken since he saw his father killed five years earlier. In the Ghost of Christmas Future's flashforward, he's ultimately institutionalized for it. After the Scrooge stand-in's change of heart, the boy ultimately whispers "God bless us, everyone" at the close.
In Uncle Dad, Debs has apparently not spoken since her mother died.
In the film Amy, a young girl feigns deafness and doesn't speak after her father is killed.
In Jumanji, ever since Peter and Judy's parents died, the former stopped speaking to everyone except the latter (who apparently is the only one who knows he can talk at all).

Literature

This is the plot of the book Silent to the Bone. The character who's silenced is accused of dropping his little sister and nearly killing her. However, this seems out-of-character for him, and his best friend tries to work with him to make him speak again. In fact, it was a negligent nanny. However, I believe the boy had a crush on her, so he kept silent.
In the Larry Niven story The Defenseless Dead, a brother and sister were kidnapped by organleggers. When they were released, the sister couldn't speak because of psychological trauma inflicted by the kidnappers.
In Halo: Ghosts of Onyx, Spartan-III Lucy-B091 was one of two survivors of a 300-soldier mission. At age 12. She hasn't spoken a single word in the 7+ years since - "Post-Traumatic Vocal Disarticulation", they called it.
Rosilda in Swedish author Maria Gripe's ...och de vita skuggorna i skogen (translated to Spanish, but not English).
The YA book Speak is about a teenage girl who after being raped has stopped talking almost completely, the main manifestation of her depression.
Marina in John Marsden's So Much To Tell You, who stopped talking after a disfiguring acid burn. The only time she speaks is at the end when she gives the Title Drop to her father. This is continued in the POV Sequel Take My Word For It, although she gets more dialogue after the above has occurred.
In Crysis: Legion, when Dr. Gould describes the true purpose of the Nanosuit as "gay rape on hanging flies", even the nearby wounded are stunned into silence. It Makes Sense in Context, but still...
Katniss has a short bout of this in Mockingjay after Prim dies.
In The Wrath of God by Jack Higgins (writing as James Graham), Emmett Keogh's new girlfriend — well, wife — has been mute for five years after witnessing the horrific deaths of her parents. When she finds her voice, it's to scream his name in warning that a bandit's about to attack him from behind.

Live-Action TV

The girl in Firefly, in the village that kidnaps Simon and River; River reads her mind, leading to the village wanting to burn her as a witch and the trope-naming Big Damn Heroes moment.
The hybrids which control the Cylon ships in the reimagined Battlestar Galactica appear to (at least some extent) show this phenomenon. One of the original Cylon models (Leobin aka 2) theorizes that their seemingly nonsensical rambling is actually highly complex and coded prophecy. Later, Baltar and Diana (and later still Baltar and Rosalin) interpret a random statement about 3 seeing 5 as Diana (model 3) seeing the final five. Eventually, this is proven true.
Felipe in the 1990 New World version of Zorro. In Isabel Allende's novel version, Bernardo's fate is similar, but his muteness is a choice rather than an affliction.
One of a couple of possible explanations for Ike's muteness in The Young Riders.
In an episode of I Love Lucy, Fred (a notorious penny-pincher) is completely unable to speak after realizing he wasted hundreds of dollars on a lemon.

Music

In the Rock Opera Tommy by The Who, the title character becomes psychosomatically "deaf, dumb and blind" after witnessing his father being killed by his wife's lover and then being told, "You didn't hear it / You didn't see it / You won't say nothing to no one."

Theatre

In one stage adaptation of The Turn of the Screw, Flora has stopped speaking after whatever it was that happened with the previous governess (heavily implied in this version to be sexual abuse).

Videogames

Garnet is like this for a while near the end of Final Fantasy IX. Combined with Fake Difficulty here, as she is your only healer for about two or three somewhat tricky dungeons (your other healer has been kidnapped by the Big Bad) and Garnet being mute means that her magic fails to activate every three or four turns.
Pilika in Suikoden II.
In Final Fantasy Crystal Chronicles: Ring of Fates this happens to Chelinka after the twins survive a monster attack that kills their dad. She eventually recovers from it, and is able to speak again.
Max in Shining Force and its remake Shining Force: Resurrection of the Dark Dragon

Webcomics

Haley was like this for a long time in The Order of the Stick. She wasn't actually mute, she could just only talk in cryptograms which changed every comic.

Web Originals

Nessiah lost all of his functionality and wound up an Empty Shell for quite some time as part of the long process of recovering his health and sanity in Dept Heaven Apocrypha. Although he's been improving steadily, his ability to speak has been the last to return. As of this writing, he's in speech therapy.

Real Life

Truth in Television for some on the autistic spectrum, temporarily for most but not all.
Maya Angelou, who wrote I Know Why The Caged Bird Sings, spent five years of her childhood nearly mute, believing that if she spoke, her words would be responsible for someone getting killed. This came about due to a traumatic incident when she was eight, when she was sexually abused and raped by her mother's boyfriend and she confessed what had happened to her brother, who told the rest of her family. The boyfriend spent one day in jail, but was found beaten to death four days after his release. Angelou credits her teacher Bertha Flowers for helping her to speak again, as well as introducing her to the classics of literature that would later serve as the foundation for her later career as a writer, poet, actress and civil rights activist.
Torey Hayden's biographical book Murphy's Boy (later adapted into the film Trapped in Silence) is based on a real-life case of this.