In Forrest Gump during the live broadcast of the first moon landing, everyone at the Army hospital is watching Forrest play Ping-Pong with himself and no one is watching the TV.
Early on in Spy Kids, Carmen and Juni are chased through a city and into a clothing store while wearing jetpacks. Only a young girl seems to notice them despite all the chaos that ensues.
The live action Super Mario Bros. movie featured a scene in which Bowser turned a Jerk Ass lawyer into an ape using a devolution gun. This happens in front of a large crowd of people who all react with laughter at the chimp in a suit scratching his butt. It seems like watching a man turn into an ape should have a much more panicky reaction. The people do start panicking when they start seeing Dinohattan replace Manhattan, however.
The whole of the film version of Uzumaki. Neighborhoods are continuously deserted, students stand head down in the hallway for an extended period in a line for no apparent reason, and it's all treated as normal. By the end of it you want the main character to get it in the head for being so oblivious.
The original Ghostbusters had a brief shot of the skyscraper with the vortex cloud rotating about it, with passersby shown walking past (in front of the camera) oblivious to the scene. This was a Special Effects Failure. (Then again, given that both this film and the sequel take place in New York City, the "Watsonian" explanation would be that the residents are so self-absorbed and/or caught up in their own misery that they don't see that one of their buildings is now the portal to a hell dimension.)
When Vinz Clortho corners Louis, the tax accountant puts his back to the glass frontage of a fancy restaurant and screams. While the diners inside may be unable to see the demon-dog to judge by his absence from the interior-POV shot, they can definitely see Louis himself freaking out. They look up for a second or two, then all calmly resume eating.
The famous part in Ghostbusters II when the RMS Titanic pulls in and the ghostly passengers walk onto the dock. One dockworker looks shocked at the happening, while another (Cheech Marin in a cameo) just shrugs and says "Well, better late than never."
Although they're walking, talking, singing chipmunks, which is apparently highly unusual in this universe given Dave's initial reaction to them, the title characters in the Alvin and the Chipmunks feature films don't seem to garner any unusual attention from anybody. Even biologists aren't knocking down Dave's door to get a good look at them!
Referenced in one of the newer movies, where one guy mentions that the success will never die down even after the chipmunks sabotaged the show because they're chipmunks that sing and dance.
Kill Bill: The main character carries a katana at all times. Even on an airliner. No one seems to object to this. Although there seemed to be a sword hilt sticking up next to every single seat. People just like to keep their swords handy in this world, it seems.
In the first Blade film, Blade walks around in broad daylight wearing goofy hair, a leather duster, and a katana. At one point he beats up a cop on the street and no one seems to notice.
Mr Barron in Carry On Doctor has prenatal depression because of his wife's pregnancy complications, so he is assigned prenatal classes with pregnant women, who don't look fazed when he's doing yoga alongside them.
One-upped in Blade: Trinity, where Blade and his two sidekicks all walk around in broad daylight carrying weapons. Even more ridiculous is that by this point, Blade is wanted by the FBI.
In Pootie Tang, the title character's father is mauled to death by a gorilla. It's certainly treated as a tragedy, but apparently it's a commonplace work hazard. At a factory. In Philadelphia.
Parodied in Shaun of the Dead, where the title character wanders around town going through his daily routine without realizing that everyone around him is a zombie. Although there wasn't that much difference between the zombies and the way his neighbors looked normally, as pointed out during the opening credits, and during the scene in question Shaun was heavily hungover. The comic tie-in has him in the midst of depression as every normal aspect of his life has gone right to hell.
In A Hard Day's Night, three out of four Beatles note that there's "a little old man in the cupboard", in his underwear. They then proceed to sit down and read their fanmail.
In Help!, no one in the Indian restaurant takes much notice of a gang of cultists strangling, kidnapping, and replacing the musicians.
Kay uses this trope to convince the newly-recruited Jay in that there really are aliens on Earth, by offering him a cup of coffee. In the breakroom, the five Worms are sitting on the counter by the coffeepot. Kay simply asks them if there's any cream, because he hates powdered coffee whitener.
Also in the first film, after Jay helps deliver a betentacled alien baby that vomits on him, Kay asks, "Anything about that seem unusual to you?". Kay is more concerned about the fact that they were in such a hurry, not the whole experience being new and strange to new recruit Jay.
The entire point is that all the weird stuff in New York City is due to them being aliens, not weird humans, as Jay realizes when taking a moment to really look at NYC at night. No one in NYC notices because they're used to it - which makes it a perfect "Casablanca" for aliens.
Kay: Most of them, they're just ordinary people, with lives, with jobs. Jay: ...cab drivers. Kay: Not as many as you'd think.
It had this situation vice versa. While agent Jay tried to convince now postal worker and brain bleached Kay that there were aliens all around them, he proceeds to do strange things with the other postal workers, who were all aliens. Upon opening the letter sorting machine, exposing a smoking, six-armed thin dude sorting all the letters, Kay simply says "no smoking" and removes his cigarette.
The sequel also had a scene where a giant alien worm went straight past a Subway station full of people, who immediately looked back down to what they were reading when it passed.
There's also two people flagging taxi's reaction to Tokka and Rahzar:
Man: Look, honey, there's a couple of monsters destroying things! What do we do if they come near us?
Woman: Let them get their own cab.
Speaking of the Ninja Turtles, the infamous "Making of..." video for their "Coming Out of Our Shells" tour includes a scene of them performing on top of Radio City Music Hall. For some reason, all the people walking down below don't even seem to be paying attention to them.
In Freddie as FRO7, the fact that Freddie's car is alive, has a penchant for jumping on other cars and makes frog noises is never commented on.
In The Meteor Man, there is a scene in which the title hero is, despite being in alter ego mode, telekinetically turning a man upside down and tearing out his pockets in the middle of a crowded sidewalk, and not one single human being even notices!!
In Indiana Jones and the Kingdom of the Crystal Skull, Indy slides under a table in the library on a motorcycle, with villains in hot pursuit. The students sitting at that table get up... and stop Indiana to ask a question about the homework, then go right back to studying. Possibly justified in that, Indiana Jones being who he is, it may not be all that unusual a sight for one of his students (and who knows when they'll get another chance to ask him about assignments?).
People seem remarkably unfazed by Howard the Duck. You'd think having a large, anthropomorphic talking duck wandering around the streets of Cleveland would attract more in the way of astonishment, but apparently not. They do notice, though. "DAT'S A DUCK!"
Little Shop of Horrors has a few scenes where the trio of chorus girls were singing in plain sight. They even walk through the flower shop unnoticed twice in the movie, and managed to change clothes for the "Downtown" number 5 seconds after they finish talking to Mushnik.
In the movie The Shadow, sword wielding mongol warriors in 13th century attire don't rise anyone's eyebrows. The Big Bad of said movie... his power was the classic Weirdness Censor. In the end, when the Big Bad's telepathy is turned off (it's unclear whether because of him being knocked out or his psionic brain lobe being damaged - which was then surgically removed to strip him of his mental powers for good) the citizens notice a friggin skyscraper they didn't notice before. Yeah, his Mind Fu was THAT strong.
The Nightmare Before Christmas: This is the only explanation for the Christmas Town elves not responding to Jack's presence during his initial visit. One would think a foppish skeleton prancing like a young gazelle would be pretty hard not to notice. Especially if he's singing the whole time. And looking in everyone's windows.
In the first TV movie for The Librarian, Judson contacts Carsen through a television set in the lobby of a Far Eastern hotel. The two Asian gentlemen who are watching the TV don't bat an eye when their program is pre-empted by an English-speaking talking head that can apparently see and hear Noah Wyle via the screen.
Averted where it counts in the Back to the Future series, with characters noting Marty's odd clothing "Why are you wearing a life preserver?" and "'Nee-kay' [Nike], what is that - some sort of indjun word?"
However the trope plays straight in the Indians' reaction to the Delorean when Marty arrives in 1885. True, they are being chased by the US Cavalry and one of the Indians hits the Delorean with an arrow, but you would think at least a few of them would stop dead.
In The Blues Brothers nobody ever comments on the song and dance numbers nor the explosions and destruction caused by Jake's girlfriend. Not even the policemen in the exploding building find it remarkable afterwards.
In the 1958 film version of Tom Thumb, the only two people in the entire village who think that an inch-high man is at all odd or worthy of some note are the bad guys, who can think only of how they could exploit him in burglaries.
In Kick-Ass, The Dragon picks up a bazooka from one of Big Daddy's safehouses, since it's a cool weapon and he's always wanted one. He's still got it when he returns to his boss. The boss calls attention to it, there's a Beat, then he's just like "ok".
Played With in Se7en. Late in the film, John Doe takes a cab from one place (Mills' apartment - not revealed until later) to a police station, and follows Mills and Somerset into the main foyer while his shirt and hands are covered in blood. Almost no one notices him or what he's doing until he stands in the middle of the foyer (and in the open for several seconds) and screams at the detectives, at which point almost every officer grabs their weapon and points it at him. If you look closely, you can see the cop at the start of the clip looking straight at him, presumably assuming he is coming in to get help. That still doesn't explain, oh, everyone else.
In Hell Up In Harlem, Tommy Gibbs is shot by a Mafia gunman in broad daylight in the middle of the street. No one stops to see what's going on, nor does anyone bother to remark upon someone brandishing a gun in a crowd of people. Tommy makes a run for it, and manages to make his way to Times Square, where (upon seeing the gunman trying to escape) uses a belt to strangle him in the middle of the square. Several people (including a father and daughter who stare straight at Tommy) don't react at all to what's going on.
In The Avengers (2012), there is a janitor whose only reaction to seeing The Incredible Hulk crash through the ceiling and create a crater, then finding a naked man in said crater, is to give said man pants and try to figure out the precise nature of the Hulk-to-man thing (after learning the man wasn't an alien, he concluded that the man simply "had a condition" and that was that).
In all fairness, anyone who watches the news in the Marvel Universe probably knows better than to antagonize the Hulk if he is near you in human form.
In the sequel, Age of Ultron, during the party in Stark/Avengers Tower, Stan Lee's reaction to Thor telling him that Asgardian liquor is too much for a mere mortal to handle, is that "neither was Omaha Beach". Stan doesn't even seem phased by the fact that he's in the presence of an actual God and a super-soldier from WW2 - it would seem, by this point, that the novelty of the Avengers has worn off a bit.
Taken even further in  - when "Bad-Ass" (Falcon) tells "Stupid Fine Writer Chick" that he's looking for a new superhero with "bomb moves", her response is that he's gonna have to be more specific, given that there's been a seeming rise in superheroes after the Avengers were formed.
In Pinocchio, there wasn't anybody who noticed that there is a giant fox and cat walking about. And yet they're the ones amazed by a live puppet without strings.
Used in Ted. At the start of a movie, a boy makes a wish as a shooting star goes by and his teddy bear, Ted, comes to life. His parents initially freak out, and for a period of time, Ted is a celebrity. As time goes by, though, people just sort of treat a living teddy bear walking around as a normal thing, though a scene where some girls ask to take a picture with him indicates he still holds some celebrity status.
In Star Trek IV: The Voyage Home, no one from 1980s San Francisco bats an eye at the Enterprise crew wandering around in their Star Fleet uniforms. Apparently this was Truth in Television: the film crew tested people's reactions by walking around in the uniforms prior to filming, and none of the residents of San Francisco noticed or commented on them.
In Last Action Hero, the Genre Savvy action-movie villain escapes from the movie into the Real World. He notices a much higher tolerance for crime in this world, and decides to test it by shooting some random guy on the street. He waits for the police cars to inevitably show up... when they don't, he shouts out "I just killed a man and want to confess!" The only response he gets is someone shouting back at him to shut up.
Played with in R.I.P.D.. Nick/old Chinese guy gets crushed by a car and a group of bystanders rush to help him. The Dead-o's actions do not go unnoticed by the media.
Splash. It's quite an endorsement of the greatness of the Statue of Liberty that a beautiful naked woman could be walking around and nobody notices her until the tour guide brings her to their attention.
No one even seems to take any notice of Noel's bloody corpse on top of the elevator near the end of Byzantium.
Mary Poppins: Every day, Admiral Boom goes up to the roof of his house where he's built a model ship and fires the (fully functional) cannons. He's so punctual that his neighbors the Banks set their clocks by him and take nonchalant countermeasures to the shockwaves.
The Rover: At the start of the film, one of the bad guys flips over their car, and it goes tumbling past the window of the bar where Eric is enjoying a drink...and he doesn't even notice.
In Paddington, nobody gives a flying shit that there is a talking bear, in fact, a very rare bear, in London. In Peru, there's a real retirement home for them.
In Short Cuts, when Betty Weathers and her son Chad return from a weekend away with Betty's boyfriend Wally to find that Chad's father Stormy (whom Betty is in the middle of divorcing) has completely destroyed every piece of furniture in the house with chainsaws, sledgehammers, scissors, and other power tools except for his mother's grandfather clock, Chad is far more interested in the business card for a vacuum cleaner salesman left in his (also untouched) Tinkertoy model and the episode of Captain Planet and the Planeteers on television (the fact that the wood casing of the television has been smashed off also escapes his notice; all that matters to Chad is that it still works).
In Rock & Rule, the audience at the rock concert apparently doesn't think there's anything strange about Mok's new singer appearing on stage in some kind of bondage device. (She's his prisoner - there's also a device around her throat that somehow forces her to sing.)
Weekend at Bernie's II: No one in town seems to care when Those Two Bad Guys shove Bernie Lomax's body into the trunk of their car, or when Richard and Larry are dragging him down the street.
Averted with Bernie's voodoodancing. Not only do Larry and Richard freak out when they first see it, but Henry and Charles - the ones who put the spell on him - do as well. Even the couple who didn't know he was a dead body were confused when Bernie got up unfazed every time the male punched him to the ground.
My Little Pony: Equestria Girls – Friendship Games: As per usual, the inhabitants of the human world are not too amazed by the Urban Fantasy-worthy goings-on. So much so that when portals open up and giant plants nearly eat two participants in the games, the complaint is that it appears to be cheating, rather than the near death experience.
Deadpool: A man in an office watches in an unconcerned manner while Deadpool beats his coworker to death with a cabinet door.
X-Men: Apocalypse: Nobody bats an eye at Nightcrawler's blue-skinned, devil-like appearance when he comes out of the theater with Jean, Scott, and Jubilee, even though many other people clearly saw him when they walked by. Kurt Wagner makes no effort to hide his appearance in the first place. This shows that society was beginning to accept mutants.