Mai-Otome: Put the Otome against the HiME, or, more specifically, their resurrected Mai-HiME counterparts.
Gundam Build Fighters features several models from different series, many of which are modified, fighting it out. So you have situations like the Build Strike vs. the GX Maoh. Heck, the initial episode where these two "fight" has the builders dressed up as Kira and Garrod respectively.
Batman and Dracula: Red Rain: finally put to rest an age-old debate: The Dark Knight or the Prince of Darkness? Dracula and Batman fight over Gotham City to find out who the true master of fear and deception is. They both lose. Dracula is killed and Batman becomes an irredeemable vampire.
Dracula vs. Superman: By hypnotizing Lois, Dracula actually gained the upper hand and succeeded in drinking Superman's blood. However, Superman is solar powered, and seconds after drinking his blood well... imagine eating a grenade.
Then there's one in Superman/Batman issue 78, which handled the subject masterfully. The fight happens, but suddenly you see a narration, or rather, two narrations arguing against each other. Eventually, it is shown that two kids, one a fan of Superman and the other a fan of Batman, arguing who wins. After much Ass Pulls done with each argument, the two kids agreed that the two are friends and would not fight each other.
Towards the end, we see Superman and Batman observing the kids. Superman wonders who would win and asks Batman on what he thinks, but Batman just flies away. Batman believes Superman will win.
"Shark vs. Train": Parodied in the children's book which takes the more common sense approach, with hilarious results (it matters, for instance, whether the competition is under water, or involves trying to sell lemonade.)
It also makes clear that Zeus would win with one arm behind the back if he decided to join the frame (so he claims, and the gods, speaking from experience, don't dare to contradict him)... And that's why he stays out of it.
And how about David Weber's Out of The Dark and vampires versus aliens? And that's just the first part of the trilogy!
Live Action TV
In How I Met Your Mother, Barney, Marshall and Lily go to the event "Wrestlers Vs. Robots" where Mexican-style wrestlers battle a variety of robots.
Deadliest Warrior: A Spike TV show which explores "history's greatest killing machines", look at their historic backgrounds and the science behind their equipment and fighting styles, and pair them off in a one-on-one computer simulated fight. Showdowns include "Gladiator vs. Apache", "Ninja vs. Spartan", "Taliban vs. IRA", "Yakuza vs. Mafia", and "William Wallace vs. Shaka Zulu".
Jurassic Fight Club: A History Channel show, similar to Deadliest Warrior in that it uses best available knowledge to establish the abilities, strengths, and weaknesses of the animals, then CGIs the actual fights. Match-ups include Allosaurus vs. Ceratosaurus; Mega-Lion vs. the Short-Faced Bear; and Megalodon vs. Brygmophyseter, the "biting sperm whale".
Animal Planet had a short lived series Animal Face Off, which was basically Jurassic Fight Club done with living species that plausibly might confront one another in the wild.
Harry Hill's TV Burp: "Well, I like X, but then I also like Y. But which is better?" [Costumed/made-up/bewigged representations/impersonators of X and Y crash into studio from opposite directions and make a beeline for each other] "FIIIIIIIIIIIGHT!!"
The original series episode "The Mind Robber" eventually, through convoluted means, features a sword fight that involved Cyrano de Bergerac, D'Artagnan, Blackbeard, and Sir Lancelot.
And "Doomsday" brings the fight people had been wanting for decades; Daleks vs Cybermen.
The novel "All-Consuming Fire" had Sherlock Holmes and the seventh Doctor square off with the Cthulhu Mythos.
Saturday Night Live: Spoofed in a recurring sketch in which George Wendt leads a cadre of football fans who speculate on the scores of hypothetical games pitting the Chicago Bears against other teams, with the Bears facing increasingly unlikely handicaps, such as being shrunk to the size of midgets. (Hint: Da Bears win.)
Wendt: Who would win, the Death Star or the Enterprise?
Star Trek had an episode in which aliens set up an Ultimate Showdown: Kirk, Spock, Abraham Lincoln, and Surak versus Colonel Green, Kahless, Zora, and Genghis Khan.
He's been stabbing Renly Baratheon for years, and Renly ain't dead!
Buffy the Vampire Slayer had an episode called "Buffy vs. Dracula". The titular fight ends inconclusively, with Buffy killing the Count, but with him clearly able to come Back from the Dead at will. The show also had to break many of its own rules about how its vampires work* though season 8 reveals his powers are a result of his skill in magic. in order to pit Buffy against a recognizable version of Dracula; the whole episode relied heavily on Rule of Cool.
This is the purpose of the World Wrestling League's Dream Matches Tour. It gets together wrestlers working in different promotions and countries who may otherwise never fight or team up, or at least may not do so anywhere where the target audience can see it.
The promotional war and invasion angles can end up like this trope, in theory. In practice, they tend to be disappointments one owner has to acknowledge to fans that another organization is at least worth comparing to their own, and as such they usually end up more like fake crossovers. The most famous flop among fans in the USA was WCW invades the WWF, an invasion angle In Name Only. Vince's kids just aligned a bunch of WWF wrestlers, some of whom happened to have gained some prior fame elsewhere. Some of whom.
Dissidia: Final Fantasy: The game, and 012 pit the lead characters from the first ten Final Fantasy titles (first thirteen as of 012, and the first to a lesser extent) against each of their respective arch-foes from their respective worlds with the fate of The Multiverse at stake.
Dynasty Warriors: Gundam: And when they didn't stop with Warriors, they decided to do the same thing to Gundam. But there are few things more awesome than watching completely unrelated Gundams ripping suits by the hundreds.
Mortal Kombat vs. DC Universe: The final epic battle ends in a draw with only Superman and Raiden left standing, but the storyline does feature DC characters winning more fights than MK characters, due to their side of the story being a chapter longer. Perhaps most notable is a scene where a weakened Captain Marvel is able to defeat Scorpion, Baraka, and Shang Tsung in quick succession.
Super Smash Bros. is the Nintendo example of this trope. It pits the Nintendo characters against each other outside their respective worlds. The final boss is the Master Hand to represent this is the characters in toy form fighting in the child's imagination.
In the Half-Life mod The Specialists, most character models are from some action movie (and two from video games): Smith Agents and Morpheus, from The Matrix; Castor, from Face/Off; Agent 47, from the Hitman series; and Gordon Freeman wearing a black trench coat. This means that it is impossible to play the game without an epic battle between action icons, unless everyone is using the generic Mercenary and Seal skins.
Super Robot Wars Z has one battle where half of the cast filled with famous Humongous Mecha anime fight with another half the cast filled with famous Humongous Mecha anime. Who wins? The side you control has to win, so you can progress through the game.
In-universe example in Brothers in Arms: Road to Hill 30: Allen and Garnett start two missions by having an argument regarding who would win a fight between Batman and Superman.
Star Wars Battlefront II has this as a mode on one map. In normal gameplay, a lucky player can become a Hero Unit such as Obi-Wan or Darth Vader. Mos Eisley offers an Assault mode, which is usually only seen in Space battles. This version has every single Hero Unit duking it out on one map. Every. Single. One. Yoda, General Grievous, Ki-Adi Mundi, Boba Fett, if he or she appeared on another map, he or she'll appear here.
Anachronism: A card game that lets you pit various historical (or somewhat historical, like Robin Hood) characters against each other. For more fun, trade the character's goodies around — give Achilles Beowulf's chain mail and Miyamoto's katana.
Who Would Win?: Exactly What It Says on the Tin. Two players each get a random character, and then an event. Argue why your character would win. Santa Claus vs. Genghis Khan in synchronized swimming? Babar vs. Einstein in ping pong?
Superfight! is another card game in this vein, where each player plays a character card and two power/problem cards (one on their own character, another on an opponent's character), then debates over which creation would beat the character of that round's judge in a fight, resulting in match-ups like Mr. T with an assault rifle and riding a unicorn vs. 3 Stephen Hawkings with the Force.
The Annoying Orange: In one episode, Liam the Leprechaun forces the orange to see if he was more annoying than Fred. It appears to end in a stalemate, only for Fred to get the last laugh.
The now-defunct WWWF Grudge Match though the archives are still there, listing the more than 200 fights the site has run. The fights are formatted in that there is a proposed scenario and a very tongue-in-cheek debate, after which there will be a vote. When the loser has fallen, often hilarious comments on the result would be posted.
In the late 2000's (around 2008 or 2009) the site was revamped to include 2 vs. 2 and 3-way battles as well as reintroducing the ability to submit characters to the database, however unfortunately at the end of 2013, the site's creator and owner stated that the CBUB (and its sister site the Fantasy Powers League) would be shutting down permanently.
TGWTG Year One Brawl brought together dozens of gamers and reviewers from That Guy with the Glasses for an epic throwndown. The winner: It's a huge draw as a truce is declared when they raelize that they are all on the same team at the end of the day.
The Screamsheet has done weekly fights between a variety of characters since 2000. The fights have a loose continuity and several running jokes, such as Superman always losing and the Earth being destroyed multiple times.
The Lonely Winds forum has three sub-forums dedicated to various USoUDs in eight different leagues of various power level and compositions.
Numerous websites dedicated solely to what faction from Star Wars and Star Trek would lay the ultimate smackdown upon all the others. StarDestroyer.net is a particularly (in)famous one, and Spacebattles.com caters to all kinds of sci-fi matchups but is invariably dominated by Trek vs. Wars as well as the occasional just-for-fun curbstomp (the Orcs from The Lord of the Rings vs. modern machine guns, carpet bombing from B-52s, an orbiting Imperial Star Destroyer, and finally, the real kicker, doze udda Orkz). Both of these websites more or less grew out of the Usenet newsgroup alt.startrek.vs.starwars.
"Dream Tournament", a popular series of vote-driven fanfics on the Usenet group rec.games.video.arcade from the mid-'90s, pitted Fighting Game characters from different games against each other. The Tournament gave rise to several spinoffs, as well as many Fanon personality traits for the characters.
There was an old fanfiction Dream Tournament called the Ultimate Video Rumble, where some truly memorable fights happened both within and without the arena. Highlights include Haohmaru and Genjuro of Samurai Shodown fame clashing swords furiously, while everyone who attempts to break their battle gets tossed aside, and dysfunctional organizers and security staff who make Eddy Gordo of Tekken fame suffer after he got eliminated from the ring.
The UVR was a deliberate spiritual successor to the first three Dream Tournaments, continuing many of the fanon events and characterizations from the DT.
The golden rule staving off the inevitable Fan Boyflame wars was understood to read, "Anyone, under the right circumstances, can defeat anyone else in a fight." The only systemic exceptions to this idea were made to account for anime-style "weight restrictions."
It's worth noting that the story of how Kasumi became God and several other plot points such as Dark Sakura originated from MTCFF Beta which was a single-elimination tournament decided by votes rather than by individual authors, with such oddities as Fusion character "Nario" resulting from Naru of Battle Arena Toshinden and Mario of Super Mario Bros. tying in votes.
"Hitler continued laughing, then finally said 'Goku! You came here expecting to find a madman, but instead, you found a GOD!' Hitler had become a Super Saiyan." How's that for a Crack Fic-style showdown? See it righthere in all its Not Safe For Sanity glory. By the way, it also ships Goku x Anne Frank.
Ultimate Showdown tournaments are a common occurrence on GameFAQs' "Board 8", allowing users to debate to the most extremely geekish limits possible about who'd win. The arguing can lead to some, hmmmm, very curious victories (Apparently, The Mask can just barely beat Broly with cartoon physics, the most powerful Darkstalkers can win left and right merely based on their vague backstories alone, Voldemort can upset Nanoha's constant nuking of areas and people and the Doctor can defeat The Flash with, uhhh, timey wimey stuff. Or prep time. Or whatever).
History nerds are somewhat fond of debating Alexander the Great vs Genghis Khan. This battle actually happens in Arthur C. Clarke and Stephen Baxter's collabaorative novel, Time's Eye.
The RPG Duelling League is a site dedicating to deciding which Video Game RPG characters would win in a fight. The site is organized into 6 week long elimination tournaments (refered to as seasons) with four different Character Tiers (Light, Middle, Heavy, and Godlike). The voters who decide who wins are also strongly encouraged NOT to vote according to popularity or plot powers, but rather comparative in-battle capabilities and performance.
Whosthebitch.com was the HQ for a fun and easy game; simply match any two given personalities (or even inanimate objects) and answer the question, "Who's the bitch?" Basically pick which one would be the 'bottom' in a relationship between the two, and justify your answer with an explanation.
Sites like Comic Book Resources and Hero Chat have their own (Comic Book Rumbles and It's Clobberin' Time!, respectively), among many, many other websites. They often don't like each other. Most infamous incident was a flame war and board raid between Rumbles and Star Destroyer, after Rumbles determined Borg Cubes can destroy the Death Star. Cue Fan Wank and board raid.
The Galactic Watercooler podcast has a recurring feature called Fantasy Sci-Fi League in which participants choose teams based on specific roles (warrior, pilot, reconnaissance, etc) and place them in a scenario such as the rescue of one of the podcasters from a Klingon prison planet, and give them some basic equipment (such as Carl Sagan's Starship of the Imagination). The winner is chosen based on the most entertaining way of completing a mission.
Factpile mostly runs on this trope. It has branched out to include other topics, but is still mostly battles between fictional characters. It has also determined the outcome of the page image Superman easily wins.
Tor.com: Tor Books' website holds yearly "cage matches" pitting various fantasy characters against each other in a single-elimination tournament, with the characters' creators invited to write how they think each of their characters' battles would go.
Video Game Championship Wrestling is an ongoing web series that uses recreations of both famous and relatively obscure video game characters (as well as some from non-game-based franchises and even a few people from real life) in the ever-glitchy WWE Video Games in AI versus AI matches. It maintains a regular roster of wrestlers, so rematches where a previous winner lost and a previous loser won have happened before, such as Charles Barkley versus Vegeta, and Nappa versus Gabe Newell.
Adventure Time: A college for worms has a "Theoretical fightonomics" class. According to the professor, the winner is always "Werewolf Queen".
Invoked and Deconstructed in "Frost & Fire" where Finn tries to get Flame Princess to fight the Ice King in a Fire vs Ice themed battle. It ends catastrophically with the Ice Kingdom getting razed, The Ice King almost getting killed, and Flame Princess breaking up with Finn when she learns of the details, because he hurt her. You blew it, Finn.
This image◊, called War of the Worlds (Война миров), painted by the Russian contemporary artist Nikolai Kopeikin depicts an epic but gruesome battle between cartoon characters of mostly Western Animation and Russian Animation. Not only this picture is an example of the trope but it also stresses the rivalry between domestic and foreign animation in Russia.
Don Bradman and Babe Ruth, widely recognised as the greatest ever players in cricket and baseball, respectively, met each other in 1932. While they didn't play against each other, certainly not in any formal capacity, Bradman offered Ruth the chance to try his hand at cricket. Ruth started by trying to bat cricket-style, and struggling to lay bat on ball, before switching to his baseball stance and hitting the Australian bowlers around the park. Ruth also took Bradman to watch a Yankees game, where Bradman surprised his host by spotting a double-play.