Street Fighter II has the Guile Handcuffs glitch, which was turned into an actual attack in the video game version of the movie. There's also invisible Dhalsim, which later got turned into his Yoga Teleport.
And the glitch that turned Ryu and Ken's fireballs red instead of blue, which was deliberately incorporated into console versions and later spawned a separate move for Ryu.
And possibly the granddaddy of good bad bugs - the ability to "buffer" the end of a normal attack's animation into a special move was originally due to a programming oversight in Street Fighter II — they decided to keep it during development, and retooled the game to make use of it.
The buffering, also known as canceling, was an official feature known as 2-in-1 and by itself was nothing spectacular. But then high level players begun to find unintentional ways to make combos longer than standard 2 hit combos:
Link Combos: When one move ends fast enough to quickly combo another move without buffering. This is the reason Guile was top-tier in early days.
The Negative Edge, a bug which allows, among other things, to throw a special projectile immediately after a normal projectile, became so popular that it was coded on purpose in many Capcom fighting games.
Kara Cancel: The ability to cancel (buffer) a move without hitting your opponent, while didn't quite play any large role in Street Fighter II, it would become a important gaming mechanic in later games (Kara Throw in Street Fighter III and Roll Cancel in Capcom vs SNK 2 for example).
The 3D Mortal Kombat games are no strangers to hilarious bugs. Particularly in Deception, if you perform a fatality motion while the victim performs the Hara-Kiri motion at the same time, this happens. At 1:12, a guy punches his head off. After punching his own arms off.
At 0:58, a guy attempts to pull another guy apart, but then breaks his back, his entire upper torso exploding right off the rest of his body. Then, what's left of his body does a little victory dance.
The SNES version of Ultimate Mortal Kombat 3 is one of the buggiest fighting games ever, but the most notable is Ghost Sheeva. Sometimes, the random character select in Tournament mode will land on the default icon. This produces a small blob of pixels (sometimes blood) given the name Sheeva (Sheeva was Dummied Out of the game). Certain characters can't hurt her at all, she can perform 100% sweeping combos, and while she can be defeated (and even hit with Fatalities if she's the last opponent, though most cause the game to glitch - she even still has her Babality sprite!), it's very hard to do so without making the game lock up.
In the Playstation version of Mortal Kombat Trilogy, turn the timer off and play a two-player match. Then set the controllers down. Someone forgot to code the music to loop, because after a few minutes, it fades out and moves on to the next theme. This doesn't just include stage themes - everything in the music select will play. After the Acid Pool theme, for example, it plays the Animality, Babality, and Continue music.
Anti AI moves in Soul Calibur 3 are moves that while easy for a human to dodge and punish, stupefy the AI and are easily spammed, the enemy only dodging coincidentally (such as dodging to the side when up against the wall, something they are doing because of the wall, not the attack). Why is this a good bug? The AI blatantly cheats in Soul Calibur 3, so it is nice to finally be able to turn that around for once.
And for some reason the final boss of Chronicles of the Sword mode is completely weak to the basic horizontal swipe of one of the sword styles. It's so stupidly easy to do that it has to be a bug...
The money glitch in 3, in which you buy everything you can from the shop, and leave the shop without saving, and then start up Chronicles of the Sword. Intentionally lose the fight/mission and somehow you get all your money back. Result: infinite gold.
Similarly in Soul Calibur 4, using Yoshimitsu's Bullet Cutter attack will break the AI. The Bullet Cutter can be held, which will turn it from a normal attack into an unblockable attack. The response of the AI will be to block until it reaches its unblockable state, then try to attack. By releasing it shortly after it becomes unblockable and then quickly starting it up again, it's possible (and usually quite easy) to beat even the hardest AI without them landing a scratch on you. Take that, Tower of Lost Souls.
The 'black hole' glitch can produce some neat graphical effects until too much stuff is added; then the game crashes. By firing a Super Scope in exactly the right pattern, you can glitch it so that it has infinite ammo. Then, set up one Fox (or Falco) that is on the same team as the one with the Super Scope, and another from a different team a certain distance apart from each other, and have them both activate their Reflectors. The character with the Super Scope then starts firing rapid-fire Super Scope shots through his ally's reflector and into his opponent's reflector. The shots bounce off the first, and then off of the second, and off the first again, in a chain reaction. If the two reflectors are properly spaced, the detonations of the shots will stabilize to the point that there is a constant mass of shots without any more needing to be fired. Then, someone who is playing as Peach can produce vegetables and throw them into the blasts, causing them to get trapped and hover in a big deadly mass. Anyone that jumps into that will promptly have their damage count skyrocket to 999 from repeated hits.
By performing as poorly as possible on the first level of Adventure mode - racking up penalties for suicide, "stale moves", etc while avoiding bonuses, it's possible to finish the level with a negative score. This counts as 999,999,999 toward your total score. However, the total-for-all-characters score is a 32-bit signed value, which means as you do this with more and more characters, the total score will loop from just over 2 billion to just under negative 2 billion, and back again; doing this trick with all characters leaves the total score at around -769,000,000, which can never be corrected except by erasing the high scores in the menu.
If one manages to exploit a bug in the Name Option of the game, which registers the character selected as "0", something interesting happens. The character that goes by the number "0" is none other than Master Hand, who is fully playable in Classic Mode and Event Mode (Only mostly playable in the Vs mode, as MH doesn't have a victory pose.) If you lose, however, it doesn't matter, as MH doesn't really need a losing pose. This glitch went undiscovered for seven years.
In Super Smash Brothers Brawl, activating Jigglypuff's Final Smash between the two halves of the bridge in the Bridge of Eldin level with the right timing can cause Jigglypuff to be knocked down without completing the Final Smash, letting you play with Giga-Jiggly.
In addition, if there is an enemy Yoshi playing on the same stage, have him use his standard B on Giga-Jiggly. It'll end up in a giant egg and pop out bigger than before. You can keep doing this until Jigglypuff gets so big that it touches the "death" areas of the map and automatically dies.
When certain characters activate their Final Smash in that same spot right before the bridge comes back, the final smash will be cancelled, but the character is still zoomed in on the character just as it is when the Final Smash gets activated, and never stops being zoomed in until the match is over.
Brawl also has a glitch with the up-and-down moving platforms in the stage creator, which allows one to "teleport" from the edge of the platform up to a sloped section of ground adjacent to that platform at the top of its movement. You can even land smash attacks on opponents that are several feet above you! See this video for a demonstration.
There is also another glitch involving Sonic's Side-B spin-dash. If he does it in a Stage Builder Stage at the beginning of a slope that leads to nothing, he'll just continue spin-dashing straight to the air, presumably having something to do with the fact that the Side-B variation of the move has a little hop in the beginning, which if done close enough to a slope won't activate and he'll just roll upwards and then the floating begins. If a character gets hit by Sonic as he rolls through the air, he'll teleport above the screen and take a really long time to get down.
There's also the exploit in the stage builder that allows Game Mods to be loaded from an SD card without having to modify your console, spawning an extremely large modding community for Brawl, unprecedented for a console game.
Final Fight had a small bug that allowed you to keep enemies in a stunlock by turning the other way just as you performed the third hit in a standard combo, thus breaking out of the sequence and allowing you to start the combo all over again while your target remained stunned. This can be done over and over again to create an infinite combo that no enemy can break out of. One of Cody's super moves in Street Fighter Alpha 3 references this bug, as he uses the trick to land an extended combo on his opponent.
Final Fight 2 has a similar bug: while playing in 2-player mode, if you attack your partner using your basic attack, you're restricted from performing a combo. If you attack both your partner and an enemy in this manner, your partner will take minimal damage while the enemy is dealt the full damage as well as the possibility of getting stunlocked if you attack fast enough. See it in action here.
In the King of Fighters 2003, Jhun has a move where he stomps on the opponent's head. However, a glitch caused Jhun to be able to roll during the hit animation, even though Jhun were still in the air. This caused Jhun to float above the ground and messed up the knockback of some of his moves, causing strange and amusing results. For example, Jhun to be able to shin kick his opponent to death. In addition, Jhun could also unfloat himself by hitting his opponent with the stomp attack again, but, if he missed, it was possible to have Jhun freeze himself until time ran out or until his opponent hit him. More amusing results ensue.
The Quest Mode of Ehrgeiz has two characters that have to be leveled up separately. Even if you just decide to focus on one of the characters, he/she dies permanently just before the final boss. As the Necro Critic pointed out, instead of spending hours leveling up the second character, you could just go to the room where the first character died and repeatedly kick his/her corpse. Since the body doesn't disappear, every hit counts as a kill. Also since the character is at a high level, every time it's kicked it raises your level as soon as you leave the room.
In Godzilla: Unleashed, there is a glitch that allows you to go outside of the arena in some levels (So far, Osaka and Seattle). This is especially fun in the Seattle level, which allows you to go to a (previously inaccessible) volcano. Unfortunately, it only works with Krystalak, Godzilla 2000, and (with great difficulty) King Ghidorah).
A separate glitch allows Mothra to go outside of Monster Island.
Rocky for the PS2 has a lot of glitches. Ugly zombie-like fighters, fighters that fall through the floor, invisible fighters and a spectacularly messed-up Clubber Lang await you. Ok, it makes it impossible to play, but it's almost guaranteed to make people laugh when they see it.
Blockofighter isn't supposed to have any special attacks. It has. Did you lose your leg? Now you can jump higher! Did you lose your arm? Now you can spin on the ground faster and deadlier! Did you lose both arms? Beyblade time! Also, you can walk on air, as long as you don't stop for more than 0.5 second.
Dissidia: Final Fantasy gives us the Link Glitch, which in simple terms tricks the game into thinking certain attacks have hit when they didn't by using a projectile attack, then a second attack at just the right moment as the projectile hits the opponent. Besides allowing attacks to hit when they didn't, it also allows attacks to combo into each other in ways they normally can't. Notable instances of this glitch include Yuna chaining Impulse into Heavenly Strike, Terra chaining Meltdown into Holy Combo, and Cecil chaining Searchlight into Paladin Force. The latter example is particularly amusing as Searchlight has significantly greater range than Paladin Force, resulting in Cecil swooping halfway across the arena in a split second to attack if the glitch is pulled off successfully.
Marvel vs. Capcom 3 uses a damage scaling system so that the more hits a combo has, the less damage each hit does, and the harder it is to keep on doing it. The DHC glitch allows players to reset the damage scaling by switching characters using certain hyper combos, allowing much bigger combos than would otherwise be possible.
Dante can supposedly perform his Quicksilver from his game, and freeze the oppoenent for the rest of the match.
Ultimate Marvel vs Capcom 3 featured a glitch that allowed players to select alternate costumes they haven't bought yet. It still involves owning a costume pack already but the upside of it is that it allows you to choose Magneto's alternate costume which is still not officially available.
Another one introduced in Ultimate, known as the Kubota Escape, allows you to leave the screen for about three seconds during a random Super, then come back in and punish if the character is still in recovery frames. This is done by what can only be described as tagging in a dead character, and then tagging the original character in.
Now patched, it was possible to turn the camera angle sideways, making it face one of the fighters head on. It allowed you to see what certain attacks looked like at that angle, which revealed interesting properties of their animations, like the X-Wave in Wolverine's Fatal Claw being so 2-D it's invisible from that angle, same with the Objection!/Hold it! phrases Phoenix Wright uses, and Zero's Genmu-Zero, which ISN'T 2-D and is actually slanted, something impossible to see from the normal angle.
There's a glitch that happens when you chip a character to death with certain supers and combo off of them on the dead character. If you perform a snapback at just the right moment before the next character comes in, you can cause both characters to come in and be used at the same time. While this does give a sort of advantage to the person performing the snapback in that it offers an easy double KO, the fun comes in how much the person who's using two characters can do, like this Haggar trick or double Level 3 hypers.
The entire competitive scene of Marvel vs. Capcom 2: New Age Of Heroes consists of these. High-level play looks absolutely nothing like what the game was originally intended to be, and a series of glitches and bugs spawned one of the fastest and most insane competitive fighters ever made. Mostly on accident.