The community-driven interactive fiction game MS Paint Adventures contains a tribute to useful bugs. Early in the game, when the player tries to pick up a gun on the protagonist's desk, it immediately turns into a key and the text parser insists that there has never been a gun in your office. When the player picks up the key, it becomes a gun again. Subsequent attempts to use the key/gun are a crapshoot as to which item it will be at the moment. Later, the player receives a number of other objects that behave the same way, and their dual properties begin to become a benefit, like a tube of lipstick which is also a chainsaw
Initially, there was a certain amount of logic to it-it would turn into a key when addressed as a gun, and a gun when addressed as a key. Later on it worked more on the basis of whatever seemed like a good idea at the time.
Lampshaded by a fake strategy guide that Problem Sleuth looks up in order to bypass a particularly hard area of the game; the key/gun switching is called "a weird glitch."
This gets referenced again during an intermission in MSPA's next adventure, Homestuck - a group of characters called the Midnight Crew use this technique to hide their various weapons as decks of cards.
This is a call-back to the pumpkin from the very first incarnation of MS Paint Adventures.
Castlevania: Symphony of the Nightactually has a community devoted to finding nifty glitches and bugs in the game. Among the ones found are ways to prevent Death from taking your Alucard Equipment, the Swords Brothers trick to go beyond 200.6% map coverage, and the Librarian glitch that gives you infinite money. Even now, after a decade, people are still finding new glitches, such as a method to get an infinite amount of Life and Heart Maxes in the Underground Caverns.
Another small exploit that's widely used in Symphony of the Night is the ability to immediately attack after landing from a jump, even if you performed an attack in mid-air just before you landed. This means you can effectively always land two swift hits on anything you encounter. Said exploit also shows up in Aria of Sorrow, to even greater effect with some of its slowest weapons.
Portrait of Ruin had an item duplication glitch that allowed you to duplicate the rewards given for completing quests, which allows the player to get the gold necessary for the Holy Claymore weapon early in the game. When combined with the Miser Ring (an item that boosts attack power based on the current amount of gold held), most enemies in the game go down pretty quickly.
It also had a glitch on the vampire sisters. You're supposed to use a purify spell that cures them of vampirism and keeps you from getting the bad ending. Unfortunately it takes around 15 seconds to charge, you can't get hit and the sisters love projectile spam. Using the "clear skies" spell that gets rid of all on screen projectile attacks causes a bug that makes them freeze in place, unable to attack. You can still damage them, so charging the purify spell is ridiculously easy.
Lament of Innocence had a place where you could duplicate gems, which could be resold for infinite cash.
Circle of the Moon had one with its DSS card system. By combining any one Action card with an Attribute card, you gained a new ability. The first card combination could be obtained very quickly; having them within 3 screens is fairly common. By itself it's not anything special. But by quickly pausing the game and switching your cards just as you activated the combo, you could use any combo in the game, as long as you knew where the cards were supposed to go on the DSS screen. You could also trigger the effects for an offensive card combination such as a summon or one of the special weapons activated with the same button combination, and before the said attacks actually go off, switch to yet another card combination that boosts your strenght to make them even stronger.
There's also a very handy little trick that works in the Bonus Dungeon. The main difficulty of said Bonus Dungeon comes from the fact that your MP is 0 throughout, meaning you can't use the aforementioned DSS system at all. But due to the way the game implements this, if you use an MP-healing item and then very quickly activate DSS, you'll have approximately 1 second in which to use your combination of choice. 1 second doesn't sound like much, but when that's all it takes to use Uranus + Thunderbird (A screen-wide nuke)...
Order Of Ecclesia has the Death Ring exploit -- A ring which gives one insane stat boosts, doubling or even tripling one's killing power. However, you also turn into a One-Hit-Point Wonder. This is usually not in the best interests of the player, except for an exploit -- Monsters summoned while wearing two Death Rings retain the stats even after said rings are removed. In other words, the 3 guards summoned by Fidelis Alate have the potential to deal more damage than you do.
In Uplink, it is possible to get a "track down the IP of the hacker who brought down our mainframe" mission in which you are the hacker. If this happens, you have an incredibly easy mission as you don't even need to use any tools: all you need to do is report 127.0.0.1 . This stunt is treated as perfectly normal by the game and you suffer no consequences from it.
Fixed in a later patch, so now it says "We cannot allow you to accept this mission, on the specific instructions of the employer" when you try and take the mission from the Uplink Internal Services Machine.
The Japanese-only NES game based on Samurai Pizza Cats has pretty tough bosses; that is, unless you use the main character's Level 3 Ninpo attack immediately as you enter the boss chamber. For some reason this causes the boss to immediately die before even entering the chamber...
The freeware game Knytt Stories has one that appears mainly in badly-made levels; when moving to another screen, if the space you're moving into is solid, you end up inside the land and slowly rising until you're on top of whatever you were inside of. Useful at times for circumventing otherwise difficult obstacles.
Egosoft's X game series has always been known for having glitches, mostly because the AI is pathetically unable to deal with fast ships. Sectors you're not in have collision detection disabled, but the problem still exists in your current sector.
In X3: The Threat there are two superfast prototype ships that the player can acquire and fly. Letting them be piloted by the AI in a sector you're in is, needless to say, the best way to have them crash into asteroids and stations.
Same thing goes for capital ships. They are much slower, but they're also much less agile and much bigger. This results in the AI being unable to efficiently calculate trajectories that avoid stations and asteroids, with predictably expensive results.
This gave rise to an interesting tactic in the older X-Tension. Xenon destroyer ships in that game were unusual in that, despite being large capital ships, they were also extremely fast - but their agility still sucked. So a player wanting to rid the universe of some Xenon ships would park himself behind a Xenon station, putting it between himself and a destroyer. Said destroyer would come at the player engines screaming, notice the obstacle too late, fail to pull out of the maneuver and piledrive into the station. This would usually result in a dead destroyer and a station with downed shields. One could then wait for the shields to recharge and repeat the procedure, or cause a second destroyer collision, which would kill both the ship and the station. It's worth noting that the destroyer would bump the station a fair distance due to its sheer mass, so it was a bad idea to park one's ship directly behind it; X-Tension is probably the only game ever to allow for the player to die from being rammed by a space station.
Up to X3: Reunion, missiles had similar problems. Specifically, it was a joke to avoid them if the player's ship was equipped with a strafe drive extension (which allows the ship to move laterally). The AI got confused if the target was moving laterally as well as forward, and the missile would never hit. This was finally fixed in X3: Terran Conflict.
Ships that have no good excuse for abnormal behaviour also occasionally smash themselves on clearly visible obstacles for no apparent reason whatsoever.
The game has a device that makes everything up to ten times as fast when it's activated (necessary to avoid terminal boredom during long space trips in slow ships). Consider the problems examined so far, and think of what happens when the game gets ran at ten times that speed. Merely being in a sector when the time compressor is activated can cause the entire local population of capital ships and large freighters to commit station suicide after smashing smaller freighters out of the way. NPC ones get recreated by the game engine, but... let's just say you don't want to go into time-compression mode in sectors where any moderately fast capital ships belonging to you reside.
In the Wisdom Tree game Bible Adventures, the David and Goliath game gives you squirrels that accidentally knock out enemies and themselves with acorns, and will occasionally run off and... fly. The Angry Video Game Nerd showcases it here.
On the Atari 2600, people used to have fun with "on-and-off"'s - glitches induced by rapidly jiggling the power switch, some of which were quite entertaining. Examples of such include:
Asteroids: the Satellite and UFO (the two enemy ships that appear occasionally going from one side of the screen to the other at various heights, either moving straight, or diagonally while bouncing off the top and bottom edges of the screen, all while firing in random directions), will only move along the top edge of the screen. The Satellite (the bigger of the two) is funny that it only fires upwards, resulting in its shots seemingly coming from beyond the bottom of the screen.
Combat: Jet fighter battles in Tank Fields, or with solid clouds (i.e. the plane would bump the clouds instead of pass through them).
Indy 500: Player 1's car becoming much bigger than Player 2's car, and occasionally, Player2's car turned into a formation of 3 cars.
Adventure: All objects randomly facing backwards, until the abovementioned "many objects" glitch makes them flip back and forth rapidly. Dragons would appear to be spinning around drunkenly, even when dead.
Missile Command: The cities turned into objects that continuously change shape, all just random gibberish, save the one at the leftmost being the most distintive that it looks like recognizable objects such as numbers.
Maze Craze: The maze turned into the type that has more than one solution, such as a complete grid (so solving the maze simply meant moving straight to the right side of the screen). This has another effect of revealing the "lurking" mechanism the robbers used to navigate the maze (eg Red robbers lean on one direction - and in the grid glitch will simply go around in circles, while Blue and Green robbers go in completely random directions). At the same time the "victory" sound for solving the maze sounded different (a simpler cute oscillating tune instead of the usual random gibberish).
Space Invaders: Player 1's cannon can fire two bullets at a time, instead of just one.
In Oregon Trail II, sometimes the game totally glitches up near the end of the California Trail, such as your leader going permanently missing in a blizzard, but the wagon train continues and he still somehow makes it to his destination. These bugs can also crash the game.
In Nation States, the names of people in the daily issues are (usually) randomly generated, resulting in a recurring character referred to as "your brother" being given a female name.
In DuckTales: The Quest for Gold, one of Launchpad's possible lines of dialogue after crashing in a flying stage "The ground was harder than I thought!" can come up even if you crash into the sea. He can also say it at the end of the game, where it makes even less sense.
Dinosaur World, an Obvious BetaEdutainment Game, offers hungry Allosaurus attacking the air above felled Diplodocus, fallen Allosaurus blinking in and out of existence if you walk past them, dinosaurs randomly walking right into each other and merging into a single entity or simply strolling into cliffsides, and also lets you fly (and thus reach the secret area) if you shake the mouse around wildly while tagging dinosaurs.