The first psychic interviewed on Peter's show was actually pretty damn close. If not for the intervention of the Ghostbusters, the world would have ended when he said it did.
"Kitten, I think what I'm saying is that sometimes shit happens, someone has to deal with it, and who ya gonna call?"
Cue applause from the jury.
Each of the three Ghostbusters who the judge wanted to burn at the stake trolls him beautifully after the Scoleri brothers' ghosts appear.
Judge: You gotta do something! Egon: Why don't you just tell them you don't believe in ghosts? ... Judge: Help me! Ray: Don't talk to me, talk to my attorney. ... Peter: You're next, Bubbles.
During the fight with the Scoleri brothers, the ghosts disappear into the walls of the courtroom after a few seconds, possibly to catch the 'Busters off-guard with another attack. In response to this, Peter begins to stiffly chuckle, to which Ray begins to quite hammily cackle like a witch. Even Egon lets off a flat "Ha, ha-ha." Seconds later, the Scoleris reappear and the three heroes proceed to wrangle and trap them. This could imply that Peter, Ray, and Egon are actually using a tactic they learned between the first and second movies to egg on or tempt violent ghosts to come after them, rather than wait for a counterattack.
Whilst capturing Nunzio, the fat Scoleri Brother, Venkman uses his proton beam to lasso him like a cowboy, and he gets into character by whooping "Yahoo! Come on, fat boy - I'm gonna take you home to my private zoo!"
Possibly a stealth callback to Venkman sarcastically saying "Nice shootin', Tex" to Egon in the prior film, when Egon's own proton beam ends up obliterating a chunk of the ballroom whilst trying to catch Slimer.
When Peter, Ray, and Egon capture the Scoleri brothers and justify their business, the trio deliver these lines that emphatically say they've got their groove back:
Ray: Two in the box... Egon: ...ready to go! Peter: We be fast and... All three:they be slow! Louis:[silently] WOW!!
They then triumphantly walk outside the courtroom.
Just the fact that, shortly after discovering the ‘negative mood slime’, Egon and Ray manage to reverse engineer it and develop an equally potent positively charged variant. If there was any doubt of their brilliance, this obliterates it.
Mayor Lenny: Alright, we've got no choice. Call the Ghostbusters. Hardemeyer: Wait! Uh, now I'm-I'm sure there's another way... Mayor Lenny: Jack, I spent an hour last night in my bedroom, talking to Fiorello La Guardia, and he's been dead for forty years. Now where are the Ghostbusters? Hardemeyer: ...They're not available. Mayor Lenny: What do you mean, "they're not available"? Hardemeyer: Well, I had them committed to the psychiatric ward at Parkview Hospital. Mayor Lenny: You what? Hardemeyer: They were threatening to go to the press! I was protecting your interests! Mayor Lenny: Oh, yeah? Hardemeyer: Uh-huh! Mayor Lenny: Well, you can stop protecting my interests. You have exactly three minutes to clear out. You're FIRED! Hardemeyer: But the election! You're making a mistake, Mr. Mayor! Mayor Lenny: Harry! Remove this man from the building, and get me the Ghostbusters!
This one builds up over time. First, the second Louis sees Oscar standing on the ledge outside the window, his immediate reaction is to urgently tell Janine to call 911 right now. He may not have been a Ghostbuster yet, but he did know when to call for help. Then, after Janosz kidnaps Oscar, Louis knows when it's hit the fan: "We gotta find the guys." While the others tackle the problem of the museum, Janine gets the idea to suit Louis up, setting up the aforementioned moments.
The Ghostbusters bringing the Statue of Liberty to life with mood slime and The Power of Soul. The song starts playing, and for a few lines, nothing happens. Then strands of pink electricity start running over the slime, growing in intensity until Lady Liberty's torch bursts into real flames. Then her feet pull away from the statue's moorings, and they're off.
At 11:56, or four minutes before Vigo will completely assume control over Oscar's body, he has begun the process, and is beginning to transfer his essence from the painting. Janosz is celebrating...when slowly, the hate-charged slime barrier at the top of the museum begins to recede, revealing a bright, burning flame held by a giant green hand. As Janosz panics, the crowd outside screams with joy as the Statue of Liberty rears back with her torch—a symbol of hope and goodness for millions of people—and destroys the museum's ceiling in a single blow. When the Ghostbusters rope down into the museum, we see her face, and she looks completely badass. Do NOT mess with Lady Liberty's city.
Also, the Ghostbusters egging the statue on.
Peter: I love you when you roughhouse! Egon: Hit it, mama! Ray: Drop the hammer on him! Winston: Go! DO IT!
Dana gets a small one in the same scene. When Vigo starts sending his energy into Oscar, she tries to move forward and grab her son, only to be pushed back by the dark magic surrounding him. As soon as the Statue of Liberty appears, this magic breaks, and rather than wait, Dana runs directly into the circle, grabs Oscar (which temporarily stops Vigo), and gets him to safety.
Ray's exceptionally furious threat to Vigo when he first materializes: "HOLD IT RIGHT THERE, DEAD-HEAD! You want a baby? Go knock up some willing hellhound! Otherwise, I'm giving you to the count of three to get back in that painting where you belong! ONE!" It's especially awesome when you remember how Ray interacted with Gozer in the first film ("Uh...good evening..."). Seems like even The HeartTook a Level in Badass in the last five years.
Alternatively, while Ray may be somewhat nervous about supernatural forces intent on destroying/conquering the world, those that Would Hurt a Child are his Berserk Button, which would absolutely be a fitting character trait for The Heart.
Vigo has the Ghostbusters on the ground unable to move much less blast him and is prepared to finish the ritual with Oscar himself. Cue Peter opening up with an EPIC "The Reason You Suck" Speech that successfully gets under Vigo’s skin and delays the possession ritual for a solid minute.
Why is the delay so important? Because this is New York on New Year’s Eve. End of the world or not, the moment midnight hits, the city celebrates, which results in a massive power loss for the negatively-charged Vigo and gets the Ghostbusters back on their feet to kick his incorporeal butt.
Vigo’s last-ditch effort to possess Ray doesn’t even make the rest of the team hesitate before Winston kicks him out with the slime blower while Egon and Peter blast his spirit into itty-bitty pieces.
playing.