12:01 (1990 Oscar-nominated short film): Myron Castleman (Kurtwood Smith) is trapped for all eternity, reliving the same hour of his life. (Even suicide doesn't help, which is a common theme in these stories. It would be hilarious to find a time loop that you can break by killing yourself.) This half-hour film aired as part of Showtime's 30-Minute Movie series.
How do you break out? You try to sue the makers of Groundhog Day, only to get trapped in endless lawyers' conferences.

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12:01 (1993 TV movie): I actually saw this when it was on originally, and enjoyed it a lot. The same day repeats over and over, due to a "time bounce" caused by a particle accelerator gone wrong. Only lowly H.R. employee Barry is aware of the time trap, and he has to get close to a lovely scientist, Lisa (who keeps dying) to discover the truth. Finally, it turns out that the main scientist behind the project (Martin Landau) is ebil.
How do you break out? By disposing of Landau, basically.

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"Endless Eight" in The Rampage of Haruhi Suzumiya: In this Japanese novel series, schoolgirl Haruhi Suzumiya can change reality. In one installment, she makes her friends, known as the SOS Brigade, relive the same two weeks over again because they were so perfect.
How do you break out? After 15,498 repetitions (or about 600 years) Haruhi finally has enough.

Star Trek: The Next Generation, "Cause and Effect": The U.S.S. Enterprise gets caught in a loop where roughly the same day repeats over and over again – always ending with the ship blowing up. For the Enterprise, the loop lasts 17 days. For the USS Bozeman, the other ship in the loop, it lasts 80 years.
How do you break out? Finally, Data is able to "remember" enough to know that Riker isn't always a total moron. Just usually.

X-Files, "Monday": It's one of those days when everything goes wrong - and worst of all, it keeps repeating. Only one woman is aware the entire world is stuck in a time loop.
How do you break out? Mulder pulls a Data, and finds a way to send a reminder to himself, so he'll do things differently the next time around.

Buffy the Vampire Slayer, "Life Serial": Buffy's working at the Magic Box, where a customer orders a live mummy hand. Her attempts to retrieve the hand invariably end with her killing it, which means the sale can't go through. How do you break out? She finally solves the puzzle by ordering a hand be delivered.

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Stargate SG-1, "Window of Opportunity": O'Neill and Teal'c get stuck in a time loop for three months, and the only way they can get out is to translate the Ancient writing on an altar so that they have the necessary information to break out of the time loop. In the meantime, O'Neill and Teal'c enjoy their freedom from consequences by biking through the base, playing golf through a Stargate, and, in O'Neill's case, resigning from the Air Force so he can kiss Major Carter.
How do you break out? They finally translate the writing, which explains how the Ancients tried to prevent a disaster by building a time machine.

Day Break: This short-lived series starring Taye Diggs had him framed for murder and reliving the same day over and over until he could solve everything. At least one other person appears to be experiencing the same repetition. Digg's character, Detective Brett Hopper, seems to be able to cause some slight changes to the next loop, such as making people wake up with a strange sense of foreboding that something is about to happen.
How do you break out? The show got canceled before we saw it happen.

Lois and Clark: The New Adventures of Superman, "Twas the Night before Mxymas": Mr. Mxyzptlk makes Christmas Eve repeat endlessly, each time around making things worse and worse until, before long, Clark and Lois (the only two aware of what's going on) are staring at nuclear annihilation as Mr. Mxyzptlk slowly removes all hope from humanity.
How do you break out? Lois and Clark finally manage to banish Mxy back to the fifth dimension, restoring hope.

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Supernatural, "Mystery Spot": Such a classic example of the Groundhog Day trope that Dean actually references Groundhog Day several times. Sam keeps reliving the same day over and over again, and it always ends with Dean dying. At first he thinks it's due to their visit to a Mystery Spot, where the laws of physics don't apply, but as Dean's deaths get more and more demented, Sam starts to suspect otherwise.
How do you break out? Sam eventually figures out the reality-warping Trickster is behind it all.

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Blood Ties, "5:55": This Canadian series about a cop with failing eyesight who becomes a P.I. (and teams up with the illegitimate vampire son of Henry VIII) featured an episode where the cop in question, Vicki Nelson, keeps reliving the same day over and over, in search of an antique.
How do you break out? Vicki finally opens a box that releases her from the time loop.

Outer Limits, "Déjà vu": A teleportation experiment goes wrong and sends a bunch of scientists back to the day before, with only Dr. Mark Crest remembering what happened. Trying to figure out what went wrong and created the loop, he ultimately discovers his friend Lt. Colonel Lester Glade altered the experiment by turning it into a bomb in the hopes of saving his career.
How do you break out? Crest finally traps Glade in a time-loop of his final second of life.

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Doctor Who, "Meglos": The British time-travel comedy soap featured a lot of time loops, but none quite as ridiculous as Meglos, where the Doctor and Romana relive the same five minutes or so over and over again, at the hands (??) of an evil cactus, who wants to impersonate the Doctor.
How do you break out? It makes almost no sense. They start reenacting the events of the time loop a minute early, so the space/time vortex gets confused. Or something. I know!

Torchwood: Captains Jack and John spent five years in a two-week timeloop.
How do you break out? Probably by having so much sex, they broke time and space.

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Sealab 2021, "Lost in Time": In a parody of the TNG episode, Quinn and Stormy keep getting blown back 15 minutes by an explosion that destroys Sealab. And when they try to warn Captain Murphy, they keep getting mistaken for doppelgangers and thrown in the brig.
How do you break out? They finally contact their other selves and abort the mission (to steal cable TV) that led to Sealab blowing up. And then all the duplicate Stormies and Quinns have to fight each other, gladiator style.

Justice League Unlimited, "The Once and Future Thing": Batman defeats the time-traveling Chronos by forcing him to trap himself in a time loop of the very second where he first started time-traveling, forcing Chronos and his wife to bicker for all eternity.
How do you break out? If Batman traps you, you stay trapped.

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The Twilight Zone – "Shadow Play": Criminal Adam Grant is stuck in an eternal time loop, as he prepares for his execution. He desperately tries to convince people that none of this is real and that it will all fade to nothing once they execute him, but to no avail: he's killed before the governor's stay of execution can come through. It all restarts with a new itineration of the loop, although this time different people pay different roles in Grant's nightmare.
How do you break out? He doesn't.

Eureka, "I Do Over": It's Allison and Nathan's wedding day, and due to some kind of wibbly wobbly timey whimey thing, Carter ends up reliving the day over and over and over again.
How do you break out? Nathan Stark finally sacrifices his own life.

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Timequake by Kurt Vonnegut: The entire world is forced to relive 1991-2001, and nobody has any free will to change events, despite remembering everything they did last time. Thus, everyone has to watch all the sadness and mistakes all over again.
How do you break out? The loop only goes around once.

Seven Days, "Come Again?": A glitch makes our time-jumping hero, Frank Parker, loop back to his arrival time.
How do you break out? Eventually, Parker figures out the glitch and prevents it from recurring.

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More lists of time loops are at Wikipedia, Me Loops, and TVTropes.

Additional reporting by Alasdair Wilkins.