Knife Fight

"Knife fight! You're gonna fight for your life!
Knife fight! You're gonna fight with a knife!
Knife fight! A really really really sharp knife!
I'm a crazy (crazy) son of a bitch,
I'm gonna cut you (cut you) swish swish

In a knife fight!"

—"Knife Fight," by Lemon Demon

Everyone likes a Sword Fight, and for good reason. It's flashy! It's suspenseful! It's Badass! It allows The Hero to prove himself not just some guy designated to defeat the villain, but who can clash with him one-on-one and come out on top (or at least hold their own long enough for them to die on their own).

Problem is, not all stories take place in locales where a sword would be considered an appropriate weapon to have along, regardless of what the hero might prefer. This might be simply because the story takes place in The Future, where no one uses swords anymore (even Laser Blades). It might also be because the protagonist is some manner of soldier, mercenary, scruffy rogue, or other manner of Anti-Hero who is thematically forbidden from using such extravagant weapons. Enter the knife—simple and useful enough that humans will almost certainly be carrying it around no matter where or when they are, yet dangerous and shiny and sharp enough that they can serve as a substitute for the Sword Fight in a pinch. Knives also have the advantage that you don't have to necessarily be a knife fighter to carry one around, meaning that any character is a candidate for pulling one out. All the drama with none of the camp!

Because knives are so small in comparison to swords (meaning that typical sword-fighting tropes can't really be used effectively with them), they tend to go a bit different from sword fights. While they can involve a lot of dodging and parrying in a sort of knive-based Flynning, most tend to be short and brutal. With knife fights, the emphasis is usually on the grit factor, with characters grunting and actively trying to go for the killing blow with every attack. Your typical Hollywood knife fight will always devolve into one character trying to stab another, with the other character grabbing the attacker's hands and trying to push them away. Unlike your typical sword fight, you can expect to see a lot of blood during a knife fight.

Common variations involve the fighters being tied to each other by the wrists (cuts down on bicycling) and both fighters being blindfolded (one participant will turn out to be a skilled blindfighter.)

Examples of Knife Fight include:

Anime and Manga

In Baccano!!, Firo has to duel Maiza before he can be accepted into the Martillo family (technically a duel, with a martial on hand who states if either of them go for the neck or torso he'll shoot them). Firon wins by slashing Maiza's arm, but of course it heals.
GitS:SAC features an unusual one—between Pazu and someone with a body identical to his own, complete with mannerisms.
Umineko no Naku Koro ni has a magical knife fight.
Not really a knife fight but in the Dark Tournament Saga of Yu Yu Hakusho the final fight of Team Urameshi's first round was settled in a Knife Edge Death Match. The idea is you put two knives into the ground opposite each other, the two fighters press up a foot against one of the knives and keep hitting each other until one of them steps off the knife.
The obscure anime Dog Soldier has one... sort of.
There was also a knife fight... sort of... during Macross's low-budget days.
Kara no Kyoukai: had a quite awesome one in the seventh movie between Ryougi and Shirazumi Lio. Take a look.

Comic Books

Storm's battle with Callisto for leadership of the Morlocks in X-Men comics.

Film

Driven to Kill features what may be the greatest knife fight ever.
The knife fight in SPL.
A fantastic one between Tommy Lee Jones and Benicio Del Toro in The Hunted.
Gangs of New York
The Bride and Vernita Green in Kill Bill Vol. 1 are in the middle of one of these when the bus comes and little Nikki arrives. They discuss continuing it at a baseball field. But in a subversion, Vernita tries for a bushwhack by pulling a gun and the Bride kills her on the spot (in front of her daughter who just walked back into the room).
Riddick gets into these a lot.
The cult film Exposure actually has a scene where a knife fight master teaches the protagonist.
Saving Private Ryan and Eastern Promises are both known for having psychologically disturbing knife fights.
The Long Riders had one between two Bowie knives, and at the lady's request no less. Notable for having the fighters "tied" together by a chew rather than their wrists.
Under Siege. Casey Ryback vs. the Big Bad William Strannix.
West Side Story
Rebel Without a Cause
Into the Night has a struggle of knives between David Bowie and Carl Perkins.
The Ultimate Warrior has a knife brawl involving no less than five people.
Avoided in Butch Cassidy and the Sundance Kid. "I don't wanna shoot with ya, Harvey!"
Say what you will about Avatar, it did manage to have the only giant mech vs. blue alien knife fight in recorded history. Also notable for one character being much larger than the other, subverting the idea of a knife fight being all about the two characters being evenly matched.
In Murders in the Rue Morgue, Dr. Mirakle's carriage runs into two men knife fighting over a prostitue.
Averted in Crocodile Dundee in the classic "That's a knife" scene. The mugger is not interested in knife fighting a guy who carries a Bowie knife around with him.
Mahoney gets into a wrist-tied version with Zed at the end of Police Academy 2. There's a twist: Mahoney has a rather small sheath knife. Zed has a scimitar.

Literature

Cormac McCarthy features several in his Historical Fiction novels. Especially The Border Trilogy and Blood Meridian, ranging from quick attacks to possibly the most epic Duel to the Death ever captured on the page.
Quite a few throughout the Dune novels; forcefield technology repells bullets, but slow things like knives and swords get through fine.
Snow Crash gives us a fight between Raven, who uses glass knives with molecule-thin cutting edges, and Uncle Enzo, who has a straight razor.
Jorge Luis Borges had a fascination for them. Knife duels feature in several of his stories ("El encuentro", "El Sur", "El fin", etc.) and there is a whole book of his poems dedicated to (a nostalgic and romanticized version of) the cuchilleros of Buenos Aires circa 1900.
Matthew Reilly does a couple of these, since most of the characters carry knives as backup weapons (see Real Life section below). Ice Station has a knife fight when the air becomes mixed with a flammable gas.
Technically its a "sword vs knife" fight, but given the relative sizes of the two combatants and the close quarters involved, the duel between Ahren Elessedil and his tormentor, Cree Bega in The Voyage of the Jerle Shannara is thematically closer to this. Oddly, given the number of Knife Nuts in Terry Brooks' writing this is the only one that really stands out, possibly due to its being between a Break the Cutie Woobie, and The Dragon.
In Ghost Story from The Dresden Files Daniel Carpenter gets into a knife fight with a street sorcerer named Aristedes, whose magic gives him Super Speed. He loses, but manages to cut him and then Butters beats him with a lead pipe before he can kill Daniel, hospitalizing him and ending his control over the others.
The Fishhawk chooses knives as the weapons for a duel against Solomon Kane, figuring that he can better use his trickery and strength than if they fought with swords or pistols.
Ship Breaker concludes with a Knife Fight between Nailer and his Archnemesis Dad, Richard. It ends when Nailer traps his dad between the gears in the ship's engine and kills him.
In Skullduggery Pleasant there are fights between tanith low, who has a BFS, and Billy-Ray Sanguine, who carries a straight razor. Hilarity ensues

Live Action TV

One episode of Magnum, P.I. in which Magnum helps reunite a couple that had been separated for decades. He was a World War II flier and she was a prostitute and they met on a beach and fell in love and planned to elope but were interrupted by the attack on Pearl Harbor (I know, I know). In any case, it shows one Flash Back seen in which the flier comes to the Place of Employment to pick up his girlfriend. The local muscleman draws a knife on him where upon a sympathetic customer throws a knife to the flier and the two of them have a Duel to the Death.
Major Zod vs Clark in the Smallville Season 9 finale. It should be noted that the knife is made of Blue Kryptonite which renders both of them human and vulnerable to being stabbed, while preventing the Kryptonian-attuned The Book of Rao from teleporting them both to another dimension. And that oh yeah, it takes place in the rain with The Matrix-style Slow-Mo being employed. The result? Clark lets Zod stab him and then jumps off the building, which results in the Book of Rao teleporting the now-Kryptonian Major away. One of the show's most epic moments.
In one of the episodes of the new Hawaii Five-O. The fight involves one of the main characters and his Broken Pedestal partner on the beach at night. Could count as a Crowning Moment of Awesome because it still has the Hollywood style flashiness but both participants were Ex-Spec Ops members and it showed off a lot of the technical aspects of a fight.
In the third episode of Stargate SG-1, Carter got in a fight with the Mongoloid leader. He had a larger curved sword and was trained to fight people his own size. Carter had a combat knife and was trained to fight people bigger than she was. Carter won.

Music

Michael Jackson's music video for "Beat It" had a switchblade fight, of the wrists-tied variety.
Lemon Demon: I'mma cut you (swish-swish) in a Knife Fight (Knife Fight), Knife Fight (Knife Fight), Knife Fight! Knife Fight!
Instrumental, but Tom Waits has the song "Knife Chase" on Blood Money.

Video Games

Leon and Krauser have one in a harrowing Press X to Not Die sequence in Resident Evil 4. It even has a bit of That One Boss to it.
In World of Warcraft, Martek the Exiled challenged Deathwing to a Knife Fight, or at least that's his story in "The Day Deathwing Came".
In Metal Gear Solid 4: Guns of the Patriots, Raiden engages Vamp in a one-on-one knife fight atop Metal Gear Rex on Shadow Moses Island. Too bad the player doesn't get to see any of it because they're trying to kill an oncoming rush of Gekkos at the same time in split-screen.

Web Original

In the Strong Bad Email personal favorites, Strong Bad claims that one of his favorite emails was the one where Coach Z and Pom-Pom had a knife fight on the stone bridge.

Western Animation

The Simpsons showed us the simple joys of a monkey knife fight.

Mr. Burns: "OH BEAUTIFUL JOE! What have they done to you?! Smithers, this monkey is going to need most of your skin…"

One episode of Drawn Together briefly featured a knife fight between look-a-likes of The Smurfs.
Parodied on the Dilbert animated series, with the office equivalent: tied at the wrist with a necktie, and fighting with marker pens.

Asok: I'm gonna mark you so bad!

Real Life

Knives are carried by most soldiers, primarily as utility/survival tools for most, while special-forces troops are more likely to use their knives as weapons. This is often reflected in the design of the knives, leading to such contrasts as the classic US Marine's K-BAR and the equally famous British Fairbairn-Sykes commando knife.
The reason Dizzy Gillespie left Cab Calloway's band involved the two of them getting into a knife-fight.
Knife fights are more likely to end in death than shooting incidents. That is because knife wounds usually occur in lethal locations, and they are usually deep and wide enough to cause fatal amount of bleeding.
The Spanish Navaja has a sophisticated martial arts culture around it.