Impromptu Tracheotomy

You know that tube between your head and your body? That thin, easily punctured tube? Yeah, guess what happens. Stabbed, flayed, barbecued, any other method to open said tube, wheezing noises optional.

Not to be confused with an actual medical tracheotomy. See also Slashed Throat.

Examples of Impromptu Tracheotomy include:

Anime and Manga

This is how Rider summons her pegasus in Fate/stay night. The mythological Pegasus sprang fully formed from the blood flowing from the decapitated Medusa's neck stump. And since Rider is Medusa...
In Black Lagoon, Yukio Washimine kills herself by piercing her own neck with a katana while Rock watches helplessly.
- This is Truth in Television, incidentally: This was the 'proper' way for a woman to commit suicide in feudal Japan because it looked more dignified than the male version (disembowelling yourself).
This is how all the murder victims in StrikerS Sound Stage X of the Magical Girl Lyrical Nanoha franchise were killed: Stab wounds to their throats via Psychic-Assisted Suicide.
In the Fullmetal Alchemist manga, and hence, Brotherhood, Heinkel pulls this on Kimblee using his fangs.
How Rika escapes torture in Higurashi no Naku Koro ni: she kills herself by repeatedly jamming her neck to a fixed-placed knife...which is bigger than her neck is wide. So messy.
Yukari in Another gets one of these near the very beginning; and from her own umbrella, no less.

Comic Books

In Elf Quest, Lord Voll gets a large crossbow arrow through his throat. Cutter, who happens to be right behind him when it exits, is only saved from a similar fate by his gold torc.

Film

Boddicker's death in RoboCop, via middle finger computer interface blade.
Red Eye: The hero stabs Cillian Murphy's character in the throat with a pen, though it doesn't stop him for long.
The Godfather Shot nastily in the throat.
- What makes it scarier is that the shot wasn't fatal, so there's a few seconds of wheezing...
In the climax of Saving Private Ryan, an American soldier is shot this way. During the ensuing melee, the audience gets what feels like upwards of half a minute of horrible choking sounds.
In the climactic fight scene of The Island, Dr. Merrick is shot through the throat with a grappling hook normally used by his guards to bring down "products."
Happens in Inside.
Saw V is a rare example of someone doing this to survive; Peter's head is trapped in a box filled with water, and he jams a pen into his trachea and breathes through it.

Live Action TV

President Hassan in Season 8 of 24.
In Sharpe's Rifles, being charged by a mounted French Trooper and not having time to load his rifle properly, Harper opts to discharge his ramrod along with the ball. The ramrod goes straight through the Trooper's throat at exactly the Adam's Apple. It is rather disturbing to watch.

Web Comics

The janitor in the first chapter of Shadownova is suddenly stabbed in the neck by Cam, even though he was bleeding to death anyway, interrupting his last words to Iris.

Web Original

In the Machinima Deus Ex Machina, a Mook is shot in the neck at one point. We get to hear this horrible gurgling sound for about 20 seconds before Deus shoots him in the face.
As one might expect, this has shown up in Survival of the Fittest quite a bit. Examples include Sally Connelly getting stabbed in the throat with a sai, and Nigel Gillespie getting stabbed with a shattered ocarina.

Video Games

And how can we forget the Assassins' preferred method for killing? Using the Hidden Blade to stab a man's neck. Ezio takes it further with his rather graphic killing of Checco Orsi, where he pushes the blade into Checco's throat slowly. In the sequel, he can cut multiple throats spinning a spear, and even jump towards a enemy and prick his throat open with a sword. It goes without saying that all of this is pretty awesome.
Speaking of stealth kills, Jensen can sneak to an enemy and do this as a melee action in Deus Ex: Human Revolution.