The Reapers in Dead Like Me often carry on conversations with the newly deceased and can calmly walk away from the scene of death without anyone noticing that it's a little bit strange. Reapers are only partly Invisible to Normals, the rest is probably just a Weirdness Censor.
Popular fan theory is that Reapers disappear when dealing with the dead.
In the Lexx episode "Mantrid," Kai's eyes keep clouding over with blackness, becoming like completely black marbles, then reverting to normal. The audience knows from the opening narration that he is possessed by Insect essence, but this goes on for quite a while without any of the characters noticing.
Similar to above, almost nobody in The X-Files ever notices that black oil is coursing through people's eyes and filling up the inside of their corneae.
The workers at the Primatech paper factory in Heroes have a tendency to ignore some slightly unusual things. Things like two random men walking into the factory unauthorized. And a passed out Japanese guy magically appearing right next to them. And one of those unauthorized intruders picking up a samurai sword that has magically appeared with the passed out Japanese guy and walking off with it. This seems to occurs every time a hero, or at least a significant character(s), enters a Mook-filled installation that's only given brief screen time.
Primatech employees aren't the only ones. At one point during Volume Three, Daphne zips out of the airport while an unpowered person is looking directly at her. His reaction to this bizarre happenstance? He blinks. None of the people on the flight with Elle and Claire seem to notice anything aside from the turbulance, as if they all routinely interact with girls who are being electrocuted without being injured by it.
Nor do any of the passengers seem to notice the aroma of lightly-fried teenage girl that would surely be around afterwards.
Any time someone uses a power in public in Heroes can fall under this trope.
In the original Power Rangers, no one (except Ernie, once) commented on the ranger's "watches." Watches with their own theme music, no less. Whenever the theme went off, those rangers present tended to snap their hands over the watches, then leave the room in one group, looking more suspicious.
In one of the many many explanations as to why it always took Tommy such a long time to appear at the scene of a Monster attack (in the Original, his counterpart Burai only had some hours left to live and had to stay in a special room where he wouldn't lose any time), He spills a drink on his Watch, causing it to go off at random times. This ends up getting his "watch" taken by a teacher, and he has to ask for it back.
Season 10 had cell phone morphers, but the identity-hiding Rangers walked around town (even college classes for Alyssa) in their color-coded, zord-logoed jackets at all times, which also had their animal catchphrase printed on it (noble lion, soaring eagle etc.) that the character would loudly yell when transforming?
"The Dull Life of a City Stockbroker" from Monty Python's Flying Circus shows the stockbroker walking to work amid gun battles, topless newsagents, etc., oblivious to all. However, all the other characters he passes by also fail to notice anything unusual happening.
Inverted in another episode where a whole town is populated by people in Superman suits, just minding their own business, until... somebody's bicycle breaks down and one of the nondescript supermen quickly changes into Bicycle Repair Man!
In Doctor Who and Torchwood, both the TARDIS and the Torchwood hub's "invisible" lift entrance have a "Perception Filter", both of which prevent people from noticing certain goings on. The perception filter on the location of the Torchwood lift is an after-effect of the TARDIS having been parked in that spot.
In some of the costumes he's worn, the Doctor himself has been an Unusually Uninteresting Sight.
While sometimes the Doctor and his companions will don period clothes before exiting the TARDIS, often they don't. Unless it's a plot point, it might only be mentioned in passing. This includes eras where modern women's fashions would be likely to get his companion arrested for being too revealing. Hand Waved in "Tooth and Claw", where Queen Vicky doesn't pay any heed to Rose's nakedness due to her giving birth to many daughters herself.
While not a sight, only once did one of the Doctor's companions in the original series ever made mention of people speaking English wherever they go. When it was mentioned by Sarah Jane, it clued in the Doctor that she'd been brainwashed because she wasn't supposed to notice.
There is also the tendency of humans to ignore unusual things, as mentioned by the Ninth Doctor to Mickey in "Boom Town".
Mickey Smith: There's no police boxes any more so doesn't it get noticed?
The Doctor: Ricky, let me tell you something about the human race. You put a mysterious blue box slap-bang in the middle of town, what do they do? Walk past it. Now stop your nagging. Let's go and explore.
Although most people are fairly shocked to see her, the villain in "The Snowmen" thinks it's more remarkable that scaly green Silurian Madame Vastra, who apparently inspired the Sherlock Holmes books or something, is a woman. Because Sherlock Holmes is a man. Never mind that he also doesn't have green scales, horns sticking out of his head, a six-foot-long tongue, and a fondness for human blood and flesh... but what really matters is that he's not a woman. Although her choice to disguise herself with a black veil is rather feminine anyway, so you'd think people would have guessed.
Charmed had a lot of those. The most common occurences are when Piper freezes the scene, and leaves while the place is still frozen. No one seems to be alarmed by the fact that a woman just disappeared in thin air.
The dozens of conversations about demons and witches in public, especially in a 'verse where their enemies could be anywhere.
Most of Piper's freezes are when people/demons are wildly trying to shoot, throw fireballs and or stab each other. Details get mixed up.
The titular character in Being Erica wanders around the streets in pyjamas and slippers kind of a lot, and nobody looks at her twice. Weird for a woman whose defining neurosis is being too hung up on what other people think about her.
This might be an example of Truth in Television as it isn't uncommon to see people out and about in their sleepwear.
Sketch show Big Train thrived off this trope. A prime example would be when a man goes to meet his friend's new girlfriend, and she turns out to be a mermaid. This is never commented on or referenced in any way.
Stargate SG-1 and Stargate Atlantis both share in this trope. Characters are often shown on missions through the Stargate openly carrying weapons at the ready into both cities as well as villages where none of the inhabitants are doing so.
I've seen every episode of both series and I can't remember a single one where this is in any way challenged by local authorities nor causing the civilian populace to flee from their presence.
To be fair, most of the missions have them interacting with pre-industrial civilizations where the locals would have no idea what their firearms actually are.
The Genii have guns, and yet they were ok with armed guests until their bunker was discovered.
Also, in the Pegasus Galaxy, gate travel is more or less routine for many planets, so they would've undoubtedly encoutnered firearms, which were independently invented by several civilizations (Genii and Satedans, to name just two). There are also the Travelers who trade with several worlds and carry energy weapons that look like regular large revolvers.
SG:1 gets challenged on having weapons pretty frequently, however, a simple "We're not here to hurt anyone" or "We're explorers" is all it takes to calm people down.
Subverted in one episode when the Stargate is being stored in a museum on a planet with a similar level of technology to Earth. They were immediately identified as terrorists and the authorities were called to deal with the 'hostage negotiations'.
Hilariously done in Community so that only observant viewers will see it. In the background, Abed can be seen having a subplot involving a pregnant woman. No one takes notice of: Abed comforting woman, Abed in a argument with woman's boyfriend/husband, the woman going into labor in front of Abed and boyfriend/husband, and finally, Abed delivering the baby in the back of a car. At the end of the episode, Shirley asks what he had been doing all week. After staring at the happy new parents in the background for a moment, he says "Not much." None of the main cast ever notices Abed's story and when Abed delivers the baby, no other background characters even bother to look.
Lampshaded later on in the season, when Shirley goes into labor during class and Abed offers to deliver the baby for her, saying he's done it before. Troy expresses surprise and then dismay that Abed "has adventures without [him]." He asks where he was when Abed did all this, and Abed says, "I dunno. Somewhere in the background."
This happens once in a while in Buffy and Angel, though not as often as people noticing weird stuff and rationalizing it or pretending they didn't.
In the Angel episode "Five by Five," Faith walks up to Angel with a crossbow, shoots a bolt at his back from 5-6 feet away, and he rapidly turns and catches it with his hand. This happens in a hotel lobby full of people who don't notice it.
In the Angel episode "Judgment," Angel and some sort of demon in full armor have a jousting duel to the death on horseback in an LA street with cars and people in the street. Nobody is shown to notice.
You see a couple of figures watching at one stage, but you'd think a few cars would stop.
Buffy famously uses an anti-tank rocket launcher to kill the Judge. Inside a shopping mall. No one seems to notice the giant explosion, or that the Judge was basically electrocuting everyone second before.
In Buffy, we get another one when Dracula visits Sunnydale. Apparently, no one had noticed until that day that there was a castle in Sunnydale.
In Are You Being Served?, Mrs. Slocombe's choices of hair color, including pink, green, purple, and others, are never remarked upon in the entire series. Considering the staff can be fired for minor things (like wrongly-folded hankerchiefs and having the wrong sort of pen in your pocket), it's especailly strange that nobody mentions it.
In Young Dracula, the Brannaghs don't seem at all fazed when the Count fires a flaming arrow with a letter attached to it into their garden table (while they are seated eating lunch at it).
In one episode of Boy Meets World, Eric gets a job on the Self-ParodyShow Within a Show "Kid Gets Acquainted With the Universe", where he meets the show's actors who are played by the other cast members of Boy Meets World, and yet he doesn't seem to notice that they look just like Cory, Shawn, Topanga and Jack.
In the UK children's show Brum, the citizens of Birmingham- sorry, the Big Town seem rather nonplussed that there's a small, talking/anthropomorphic, yellow car driving merrily around. However it could be argued that they're all just used to the sight, as Brum is often recognised and known by name to them.
Something of a meta-example from Merlin. The show is filmed in the real French castle of Pierrefonds, which features a bizarre stone statue of a pelican with exposed breasts on the balustrade of the castle's exterior staircase. Often it appears in the background of certain shots, but so far none of the characters have commented on it.
The guards don't seem to find anything strange about dice deciding to roll away on their own.
Burn Notice's season 1 finale had a hilarious real life example. The episode's Big Bad blows up a car with a shoulder-fired missile, causing an enormous ten-story fireball to erupt from the car. And drivers on the nearby overpass just keep driving. The director said on the DVD featurettes that he started to complain that the extras on the overpass missed their cue ... then he realized those were actual Miami drivers completely ignoring the explosion.
Truth in Television: The cast of Tanked lampshades this when they install a giant gumball-machine/aquarium in front of a client's business, and notice that virtually nobody on the busy street even pauses to stare at them. One of them quips that it's Las Vegas, meaning Elvis could ride down the Strip on a Seageway wearing a Santa suit and nobody would notice.
In the pilot episode of Flashpoint, Ed in full police gear and carrying a sniper rifle steps into a crowded elevator and casually asks someone to push the 10th floor button. The occupants look surprised, then amused.
Doc Martin: What is the reaction to a man holding a baby, giving a testicular exam to a policeman, in a public lavatory?