Interstate 60 has many examples of this trope, which is the subtext of the movie, until the point in which, iconically, the hero drives his car off the road instead of choosing one of two roads he doesn't want to go down.
In Catch Me If You Can, Frank Abganale Jr. has to make a choice concerning whether he wants his mother or father to have custody over him: unable to choose, he runs away, thus beginning his trail of bank fraud and crime.
The movie The Adjustment Bureau has a big one. The main character is forced to choose between having a romance with the girl he was "destined" to be with and thus dooming her career, or breaking up with her and allowing them both to become famous and influential, she as a dancer, him as a politician. While initially he chooses to break up with her, he later finds out that because of this she's going to be hugely successful but very unhappy. So he takes option number three: he tries to convince God to change destiny.
In Chariots of Fire Eric Liddell has to either give up his Olympic race, or run on Sunday (which he is opposed to do so because to him it would violate the Sabbath). So, he decides to withdraw and enter a different race instead.
In Titanic (1997), Rose is unable to get her beloved Jack since he died in a Heroic Sacrifice, but she neither wishes to marry her arranged fiancé Cal who is almost surely looking for her. She assumes a false name on RMS Carpathia after having been rescued, thus distracting Cal; she reaffirms it by looking away when Cal himself is nearby, managing to dodge him.
In The Matrix, Morpheus is captured by the Agents and Neo has the choice of letting him be forced to reveal all his secrets, which would doom the resistance, or unplugging his body from the interface, which would mean instant death for him. However, Neo refuses to make either choice and decides to go in and rescue Morpheus instead despite the formidable opposition.
In The Matrix Reloaded, Neo is told Trinity is destined to die, and he can't save her. His solution is to bring her back after she kicks it. Problem solved.
In The Matrix Revolutions, the second issue is ending the war, which seems certain to end with either the machines or humans being wiped out. In the end Neo unites them against a common enemy.
In the first Spider-Man film, The Green Goblin forces the hero to decide whether to save Mary Jane or a Roosevelt Island Tramway cabin filled with young children. With great personal effort, and some help from the public, Spider-Man saves both.
In The Cat in the Hat starring Mike Myers, while the kids argue whether to get the dog or not, the Cat claims, "There is a third option... but it involves...murder."
In Batman Forever, the Riddler forces Bats to choose between his brand-new sidekick or the girl. Using his wits and a few of his many wonderful toys, he's able to save both, defeat both villains (Two-Face flips his last coin), and secure his secret identity.
In The Dark Knight, this is both subverted and played straight. Batman saves Harvey Dent's life at the cost of Rachel's, since Commissioner Gordon would never make it in time to save her, with the added twist that the Joker purposely gave them the wrong addresses. Later on in the film, the civilians and the prisoners decide against sacrificing the other group in exchange for their lives like the Joker had originally planned.
In Superman: The Movie, Lex Luthor launches two nuclear missiles headed for opposite ends of the country so that Superman can't stop them both. By reversing the flow of time, he manages to do so anyway. This was perhaps not the fastest option, though.
In Who Am I?, Jackie Chan's character is confronted by two thugs on the roof of a building, who give him two choices: "Give us the disk and jump off," said disk being the MacGuffin of the movie, or "We take the disk and throw you off." Instead, Jackie says, "I like the third choice: I keep the disk and throw you both off." (During the ensuing fight scene, nobody actually gets thrown off.)
Subverted in The Rundown, when Beck first confronts Travis. Beck offers Travis two choices: Option A (come quietly back to LA) or option B ("Pretty much the opposite of A. But I wouldn't recommend that one.") Travis decides to go with Option C, and attempts to beat up Beck and escape; considering that Beck is the hero of the movie, and is played by The Rock, while Travis is played by Stiffler from American Pie, you can guess how that goes. Afterwards, Beck quips, "Like I said, there is no Option C."
And as Beck attempts to leave the bar with an angry, handcuffed Travis in tow, in walks Cornelius Hatcher, who refuses to let Beck leave with Travis.
Travis: Allow me to introduce you to Option C...
By the end of the movie, a true Option C presents itself: Beck takes Travis to LA, but helps him escape his father's wrath.
Subverted by The Proposition, in which outlaw Charlie Burns is given a Sadistic Choice: find and kill his older brother Arthur, or his younger brother Mikey will hang. He and Arthur attack the jail and rescue Mikey, but because of a meddling Smug Snake, Mikey dies soon after. And then Charlie ends up killing Arthur anyway, but for a different reason.
Spoofed somewhat at the end of the Peter Sellers comedy I Love You, Alice B. Toklas! Faced with the choice of confining-but-stable upper-middle-class marriage or free-spirited-but-superficial hippie life, Sellers' middle-aged hero runs away from his second attempt at marrying his fiance from the former group. When asked by a passerby where he's going, he admits he doesn't know, and he doesn't care. He is determined to find a third option - the one that will bring him happiness.
The film is a heavy deconstruction of Take a Third Option: Kirk has made a career on taking the third option, so when situations finally arise where there and definitely is no third option, he's completely unprepared.
In Star Trek IV: The Voyage Home, the Federation has a tiny problem at the end: they need to punish the crew of the Enterprise for their actions in the last film, but they can't just out and out punish them after they got through saving the Earth. The third option? Drop all charges except for one, which ends up demoting Kirk down to captain. Then, give him a new Enterprise.
In a slight subversion in Willow, the main character and two others are asked by the village's High Aldwin (town wizard/leader), holding up his hand, which finger contains the power of magic. Each chooses a finger, and the Aldwin declares that they're all wrong. Later, in private, Willow tells the old man that he had thought of choosing his ''own'' finger, but rejected the idea. The Aldwin confirms that this was, in fact, the correct answer.
In Snatch., Irish Traveller character Mickey is given two choices: perform in a fixed Forced Prize Fight that he has to lose by knockout in the fourth round, or watch as the gangsters forcing him to do this murder his entire clan. (And just to prove they will, they set fire to his mother's caravan — with her inside.) Instead, they decide on a third option. Mickey and the entire clan bet a shitload of money on Mickey knocking out his opponent, which he does, and before the gangsters can do anything about it the clan kills all the gangsters assassins and the head of the mob as well. Moral of the story: don't fuck with Pikeys.
In the first Iron Man film, Tony Stark is given a choice of building a Jericho missile or getting his head blown off. Stark instead uses the parts to build a suit of Power Armor and fight his way out of the base.
Fox Mulder did this in The X-Files: Fight the Future. When he followed the not-so-mysterious government trucks he came to a T-intersection and didn't know whether they went left or right. He then decided to drive straight ahead through a field road. Needless to say, being Fox Mulder, he was right.
Subverted in Punisher: War Zone: the villains trap Frank in a Sadistic Choice that forces him to kill one hostage to save another. Frank instead elects to take a third option, killing one hostage taker to release his captives, consequently sacrificing the other captive to the other hostage taker. However, since said hostage had already offered his life to Frank beforehand to save the others, Frank essentially made a choice that accomplished the same result, but without the hostage's blood in his hands.
In The Empire Strikes Back, Lando Calrissian is forced into a deal with Vader - he betrays his old friend so Han will be bait for a trap, and afterwards the crew will be safe on Cloud City, and the city itself will then be ignored by the Empire, which will kindly not destroy it. But the deal gets altered repeatedly. So Lando made plans to free the crew of the Falcon and get the citizens to flee, though he could only execute them after Han had been frozen in carbonite. The third option was only partially successful, but things turned out all right in the end. Even Cloud City apparently escaped harm - the Star Wars Expanded Universe tells us that it fell under Imperial control, but never got destroyed (perhaps the gas mines were too profitable), and eventually the New Republic got it back.
And in Return of the Jedi Luke has the option of either killing Vader and Palpy or joining them. What does he do? He chooses to die, and in the process inspires his father to save him, killing both Sith in the process and bringing balance to the Force.
Revenge of the Sith has an ironic echo of the above and a rare negative outcome of this trope. When given the choice between supporting Mace Windu and killing Palpatine or supporting Palpatine and killing Mace Windu, Anakin chooses to disarm Windu. This immediately leads to Mace's death by Force lightning.
In the 1999 movie Wing Commander, Maniac offers Hunter the option of either kicking his ass, or drinking the booze that Maniac brought. Rosie suggests a third option: kicking Maniac's ass, then taking his scotch.
In National Treasure, Nicolas Cage's character is given a choice by the feds; do nothing and go to prison, or help the feds rescue the stolen Declaration of Independence from Ian and still go to prison ("But you'll feel better on the inside"). Initially, Cage accepts the second option but when the plan goes south, he declares, "Sadusky, I'm still not against you. But I found door #3, and I'm taking it." He then makes his escape via diving in the Hudson river.
Occurs in the film The Red Shoes (1948). The main character Vicky is forced to choose between Julian, the man she loves, and dancing ballet, which she considers as important as living. At one point, she has a relationship with him, but keeps it a secret. When they are discovered, things get more complicated and in the end she decides to commit suicide by throwing herself in front of a moving train.
Played with in the third Pirates of the Caribbean movie, when Will is trying to decide between killing Jones in order to save his Dad, which would mean he would become the new undead captain of the Flying Dutchman or remaining alive and marrying Elizabeth, Jack proposes that he "avoid the choice entirely. Change the facts" by letting someone else kill Jones. In the end, this doesn't work out, because Jones stabs Will, and in order to save his life, Jack helps will stab Jone's heart.
In the James Bond film For Your Eyes Only, Bond and General Gogol confront each other over possession of the ATAC nuclear weapons transmitter, after killing the Big Bad, Kristatos. Bond then destroys the ATAC by throwing it over a cliff, then comments, "That's détente, comrade. You don't have it, I dont have it."
In Sherlock Holmes: A Game of Shadows, Holmes' final confrontation with Professor Moriarty over the Reichenbach Falls boils down to a choice between letting Moriarty go, knowing that Moriarty will take revenge by killing Dr. Watson and his new wife, or trying to fight the former Cambridge boxing champion while suffering from a serious shoulder injury, which he deduces will result in his own death and Moriarty going after the Watsons anyway. Holmes' third option is to grab Moriarty and throw them both over the Falls.
According to How It Should Have Ended, there was a fourth option that Holmes hadn't considered - calling Watson to even the odds. Moriarty may be a former Cambridge boxing champion, but Watson was a soldier and is also much younger. Of course, Watson was busy trying to find Moriarty's agent, and Holmes couldn't know when he would be finished or free him up. Since we know Holmes' only goal was "keep Watson safe", involving him in a fight would not feasible.
Coach Halas does this in Brian's Song when he can't decide if Brian Piccolo or Gale Sayers should start. He decides to switch Piccolo to fullback so he can have both players in the starting lineup.
In the 1985 version of Brewsters Millions Brewster's "None of the Above" political campaign results in neither of the two candidates wining thus forcing another election.
James P. "Sulley" Sullivan is left in charge of Monsters Inc with a dilemma: continue to operate the company harvesting the screams of small children, after seeing how terrified it makes them? Or let the company collapse, leaving him in poverty and shame and Monstropolis in an eternal electrical blackout? Sulley goes public with the discovery of laughter, a power source ten times more potent than screams, compatible with existing Monsters, Inc. technology, and entirely kid-friendly.
In Wargames, Joshua, a.k.a. WOPR, realizes the futility of playing "Global Thermo-Nuclear War" after utilizing every scenario and gambit in the book without altering the game's outcome. In the end, it says "A strange game. The only winning move is not to play." Amen.
In Curse of the Golden Flower, Prince Jai must either give poison to his mother and thus help to kill her or let himself be executed for treason. What's his solution? Commit suicide.
Bridget Jones, from the first movie Bridget Jones's Diary: Bridget has two options. She can either accept a permanent state of spinsterhood and eventually eating by Alsatians, or NOT. Bridget chooses vodka. And Chaka Khan.
Towards the end of The Bank, the father who blames the bank's CEO for his son's death bursts into the CEO's office with a shotgun with the intention of killing him. The CEO tells the man that while his son's death was tragic, he doesn't consider himself responsible, but to go ahead and do what he thinks he has to do. As the man contemplates whether to go through with it and spend the rest of his life in prison or forgo his revenge and possibly still do some time, the phone rings. The CEO insists on answering it. The man realises the CEO would rather die than miss the call, so he shoots the phone, then goes outside and shoots the fuse box.
In Pixar's Brave, Merida is faced with two choices. One is arranged marriage when she doesn't want to be married at all, much less to some stranger. Two is cancel the tourney and provoke war between the clans. So Merida, who is a pretty good archer on her own, decides to try some Loophole Abuse and win her own hand in the contest. Which turns out to insult the other clans just as much as a cancellation. Oh, and she turned her mother into a bear too. Which makes things worse because her mother is the person who keeps the clans in order, and without her there they cannot be appeased.
Played for laughs in Diner; Eddie is bugging his friends by asking them whom they prefer, Sinatra or Mathis. Boogie answers, "Presley." Eddie is not amused.
When Carol tells Admiral Marcus he can't destroy the Enterprise with her on-board, he simply beams her onto his ship, then prepares to re-open fire.
Khan threatens to attack the disabled Enterprise unless he gets his crew back. When Spock points out destroying the ship will also kill Khan's crew, Khan replies that he'll just destroy the life support and cut the oxygen. Since his crew are in cryosleep, they'll remain unharmed as the Enterprise crew suffocates.
Megara from Hercules takes one of these via using a Loophole Abuse. One option is to do nothing and let Hercules be defeated since he's been Brought Down to Normal through a deal with Hades that she was involved into, in exchange for her physical safety. The second is to have Hercules break the pact, which would get her hurt. The one she takes is to break the pact herself via pulling a Diving Save to keep Hercules from being crushed by a falling column; as soon as she's injured, Hercules recovers his powers and is able to fight back. (Too bad Megara's injuries are fatal, however. But Hercules himself finds a way to get around that, too).
In the climax of Big Hero 6, Hiro must encourage four of his mates to do this by fleeing from attacking microbots, since their normal attacks don't seem to work. Wasabi gets pinned down by some between slabs on concrete on either side, but he cuts into concrete in order to escape these microbots. Gogo gets encased in a microbot-orb, but starts circling inside them to burn her way out. Although Honey Lemon gets trapped in her chemical dome while trying to protect herself when it is failing, she attaches her chemicals to microbots intstead to pull her out. Lastly, Fredzilla gets his suit's limbs seized and realizes that it's like his mascot's costume. However, he pulls his arms out to grab some debris in order to free himself with.
In Kingsman: The Secret Service, Valentine's plot is this - rather than let humanity be wiped out by global warming or allowing global warming to rise to such an extent that Earth's ecosystem is destroyed and humanity dies anyway Valentine tries to wipe out nearly the entire population allowing humanity to survive and global warming to stop.
The Hunger Games: When Katniss and Peeta decide to take the poison berries to kill themselves rather than one having to kill the other.
In Tangled, Mother Gothel is about to forcefully take Rapunzel away, but Rapunzel promises she'll go with Mother Gothel willingly if she can use her hair to heal a fatally injured Flynn first. Flynn quickly takes a third option by cutting Rapunzel's hair before she can heal him, allowing Rapunzel her freedom while denying Mother Gothel from her source of eternal youth and denying himself a chance to be healed from his mortal injury. He dies, but then the Swiss Army Tears kick in and he's revived.