One Piece / OOC Is Serious Business

General

Luffy, the captain of the Straw Hat crew, is known for being a goofy and adventurous sort. However, when he drops the goofiness, you know shit is about to get real.
The same goes for the rest of his crew. In their normal state, they will often joke around with each other, goof off, or even fight each other over petty little squabbles. If they're not goofing around anymore, you know some serious shit just went down.

At one point in the Alabasta arc, Vivi says three words to Crocodile: "I'll Kill You!". Mind you, this is coming from a young princess who is such a Nice Girl that she can't stand the idea of a single person dying in a war.
When Luffy points out the very harsh, but necessary truth that it's naïve to hope that all people involved in a war will survive, Nami gets angry at Luffy for being so blunt and not considering Vivi's feelings. It's extremely telling that Sanji is the one to tell Nami to shut up and not cut off Luffy. Usually, he would never show agreement with his male crewmates if he can please his female crewmates instead, but in this situation, he apparently agreed with Luffy so much that he was willing to let it slide or thought that Vivi needed to hear it.
When Robin encounters Tashigi's squad of Marines in Alabasta, one of the Marines mentions how Robin destroyed six ships as a young girl, which is why she got a 79 million berry bounty at the age of eight. Robin, who'd been rather stoic up to this point, then becomes furious, since it later turns out that the entire story behind her bounty was nothing but a lie told to justify hunting down the only survivor of Ohara. In reality, the destroyed ships were a combination of Saul destroying some ships trying to save Robin, and Akainu sinking a ship full of civilians.

While Luffy usually believes Usopp's lies, when Usopp tries his "80 million followers" bluff on Luffy before their fight, as a Call-Back to their first meeting, Luffy bluntly says he knows Usopp doesn't have a single follower, which helps drive home how serious the fight is.
Franky's mentor Tom is generally a Nice Guy who's rather patient with Franky even when he ends up slacking off to build his "Battle Franky" battleships. Unlike Iceberg, Tom doesn't even blame Franky when Cipher Pol steals the warships to frame Tom, Iceberg and Franky for attacking the judicial ship, thereby getting them sentenced to death. However, when Franky disowns the battleships he made, Tom breaks his chains and punches Franky, saying that Franky must love his ships no matter what they end up being used for.
Robin freaks out when the Buster Call happens. For good reason: it destroyed her hometown and killed everybody who lived there.
Sanji has a repeat of his moment in Alabasta in the Post-Enies Lobby arc, when Zoro is completely adamant that Usopp not be allowed back into the crew until he apologizes for disrespecting their captain. When Nami protests that Zoro is being too harsh, Sanji actually speaks up for the first time to say that, to his own chagrin, Zoro is absolutely right. The fact that Sanji openly sides with Zoro over Nami says a lot about just how strongly he must have felt about the issue.

Played for Laughs at the end of Thriller Bark, when Nami gave treasure to her new friend Lola. Luffy and Usopp were shocked, and actually shouted "A STORM IS COMING!"

Luffy usually holds back against much weaker foes and most of the time he even respects his foes a bit. However, when he angrily knocks out Charloss in a single blow, he does neither of the above. Considering who Charloss is in the grand scheme of the One Piece story, this was also a major Oh, Crap! moment for literally every single character who witnessed the punch.
Another serious moment from Luffy is in the Sabaody arc when the crew is attacked by a massive heavy-hitting task force, consisting of Sentomaru, a Pacifista, and Admiral Kizaru(who is on par with Aokiji, who'd effortlessly defeated Luffy a few arcs ago), just after the Straw Hats had barely defeated another Pacifista. The situation was so bad that Luffy, a stock Determinator actually ordered his crew to escape the attacking Marines, lest they all be wiped out.
On a far more comedic note, when Buggy learns that his precious Buggy-Balls left absolutely no impression on Luffy, he's so dejected that he actually closes his mouth.
Luffy actually uses this on purpose to relay a secret message to his crew after they've been separated to meet up in Sabaody after two years. He appears to be performing a memorial service to those who died in the war, but it's actually so that he can be photographed with a tattoo that has "3D" crossed out and "2Y" next to it- the Straw Hats won't meet at Sabaody in three days, but rather, two years.

Corazon is thought to be mute by everyone, including his own brother. Which makes it so surprising when he finally talks to tell Law that he needs to get away from Doflamingo.
When meeting with retired Fleet Admiral Sengoku directly after Doflamingo's defeat in Dressrosa, Trafalgar Law is nothing but polite and respectful to the older man, which is in stark contrast to the rude snark and Flipping the Bird attitude he usually gives to authority figures. Turns out, being able to reminisce and share fond memories with the only other person who loved Rocinante Donquixote as much as he did is more than enough for Law to put aside their differences, at least for the time being.
Another moment was when Robin met the innocent and utterly gullible Tontattas. She's usually The Stoic, who has understated reactions to most events, but ends up having her first Wild Take.
Another moment was when Robin learned the truth about how Doflamingo came to rule Dressrosa- forcing the once beloved king to demand money from his subjects, then using his People Puppets ability to force him to attack them, then "saving" the kingdom from its king. This infuriated her so much she threw in with La Résistance, even when she's normally a pragmatic woman who won't advise an open war.

During the Whole Cake Island arc, Sanji turns down a beautiful woman he's set to marry to stay with the Straw Hat crew. Even the other Straw Hats are amazed by it.
- A much more serious example happens when Luffy vows not to eat any food unless it's prepared by Sanji, as an ultimatum to get him into the crew instead of going off to get married. Which Sanji clearly doesn't want, but he's being forced by explosive bracelets and assassins aiming at Zeff.
Another Played for Laughs example. One Running Gag in the series is that Nami would happily cook for the crew, but only if they pay her. After Luffy attempts cooking and ends up wasting all the food into an inedible stew, causing the crew to nearly starve to death, Nami steps up and says she'll cook without payment. Shocking Luffy that she'd do it for free.
Another moment was when Robin read the news about Luffy's Leeroy Jenkins against Kaido, and the resulting Curb-Stomp Battle and Luffy's subsequent imprisonment.

In Baron Omatsuri and the Secret Island, Luffy kills his enemy for once. And it really was deserved.
A moment where it was played for laughs is during a TV special: As the Straw Hats were performing a play, Robin started overacting, dancing around and singing as if she was in an opera. Understandably, Chopper was completely freaked out.