A spreadsheet typo caused one of the players in Madden 06 to be mere inches in height! Gamers enjoyed Madden for the first time in years, but the bug was quickly fixed, killing off the amusement.
John Madden Football '92 features an ambulance that retrieves injured players, which also end up running over any other player that get in the way. See the hilarity in all its 16-bit glory.
In NHL 09, if a goal ricochets into the net off a defender's stick, that defender will be shown celebrating with the other team in the goal cutscene. It's just too funny to eliminate.
It gets better. If you pokecheck the puck from the opponent's goalie into the net, he celebrates with your team. Goalies should celebrate goals more often in real life.
Another bug allowed players in the Online Team Play mode to glitch themselves to 99 Overall stats (rather than the 75-85 most had). Some actually preferred this because it sped the game up quite a bit.
The Amiga rugby game International Rugby Challenge would keep running the match timer while the game was paused. Stuart Campbell's review in Amiga Power magazine—which awarded a score of 2%, for numerous other bugs—pointed out that it was possible to win any game by scoring an initial try, and then pausing for the remainder of the match.
Along similar lines, the infamously poor Amiga football game Kick Off 96 featured a bug whereby the player could line up for free kicks, penalties etc indefinitely, whilst the match timer continued to count down. The player could exploit the exact same match-winning bug as in International Rugby Challenge, by scoring a goal early in the game, getting a goal kick, and waiting for the clock to run out.
Some of the glitches of other Kick Off games are more fondly-remembered. For one, sometimes when the ball struck a goalpost it would ricochet off with far more velocity than it previously had. This could result in a goal at the other end of the pitch!
For two, a player who falls over as part of a legal tackle, would sometimes stay on the ground but be able to move around like a normal player. Well, almost like a normal player: they would also have super speed!
A glitch in Skate 2 allows you to fling yourself into the air under the right circumstances. If you drop your board to send it rolling along angled or uneven ground, then press a set of buttons with the right timing (in the case of the 360 version, it's X to jump, then the left trigger, Y and Right on the left analog stick all at the same time), your skater will grab his board in mid-jump, and his feet will snap onto it like magnets, causing him to hit the ground body-first and bounce high into the air. It doesn't work everywhere and it won't work online, but it makes for some hilarious bails once you get it right.
In Tony Hawk's Pro Skater 2, doing many tricks rapidly in succession would cause the player to pick up speed. This gave alternative methods to get to certain areas (it is possible for instance to launch off a concrete bank in School 2 to find an alternative route to the area containing the Carlsbad gap that is usually used as the exit from that spot in the game) and it's pretty much the only simple way to do the Love gap. It can also be used in the Marseilles level to do 1080s and 1260s, or a "360 the 900".
It's also possible to rack up massive rotations if you do donuts in front of non-standard gaps such as the loop in The Bullring or the pass-through on the Enma Chen ramp at Skate Heaven then rolling through the gap.
This video shows how in Major League Baseball 2K6 a Fenway Park outfielder can prevent a home run by jumping the height of the Green Monster In a Single Bound.
Any of the various bugs that exists in Football Manager that aren't outright cheats but still give the human player an advantage if they choose to use them. The most well known are the "Corner Bug", which are when given a specific set of instructions at corner kicks, they will result in more goals than usual. Over the years they have included firing the ball out to the edge of the penalty box for a long shot, moving all but your best player away from the near post then having the corner taken there, the same but the back post.
Another well known bug is setting a player to be sold to another team with international appearance clauses. Whilst you are manager of the international team. Every so often a version of the game will come out where the AI doesn't recognise this situation, and will happily include a "play 10 games and pay 50 million pounds" clause because they see that the player they are buying isn't likely to play internationally, but doesn't realise that the human player can easily bring them into the team and play them and get the money.
Then there are the overall Game Breaker tactics. The most well known was the Diablo tactic for Championship Manager 03/04. To this day the top tactics are compared to it.
Tiger Woods PGA Tour 08 had a bug made famous by a YouTube video, where a ball landing in a water hazard could actually float on the surface and the game would count it as a playable lie. The end result would be Tiger Woods walking on water to hit the ball. EA Sports actually recreated the glitch in live action with Tiger Woods himself for a Tiger Woods 09 promo and jokingly claimed, "It's not a glitch. He's just that good."
An odd bug in the NES version of Punch Out made Soda Popinski pathetically easy. If Mac ducks when Popinski crouches to throw an uppercut, Popinski will freeze for a couple seconds. Mac can throw a body blow and usually get a star. Even stranger, this glitch will cause Soda to be knocked down after any star punch.
The C64 game International Soccer featured the glitch that you could run the length of the pitch, and score, with the ball bouncing on your head.
Skate 3 is absolutely FULL of these. There's a glitch to get your board stuck in a piece of plywood which will then lift off like a magic carpet (with you on it) and fly around the map. There's a glitch where you can get your board stuck under a manny pad while doing a set up and take off into the sky. There's a glitch that allows you to launch yourself off any ramp at high speeds and get all sorts of places the developers didn't intend for you to be. There are several pipes with the geometry reversed that your skater can get stuck inside and become a scrambled squirming mess. There's a glitch that makes you shoot off into the sky in a superdude position. There's a glitch that makes your skater explode into a rapidly flailing ball of twisted limbs. There's a glitch that makes you fall through the ground and into an endless void of darkness (or inside a building). There's a glitch where half your board gets stuck underground and pushing off causes your legs to bend and rotate in hilarious ways. There are trash cans that spawn inside of benches and go flying. And there's more. So much more. It's actually possible to identify Skate 3 just from the sound of its physics engine freaking out.