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This is such a common advertising trope that listing each occurrence would eat up an entire page of its own. In fact, one of the most basic ploys in advertising is to convince people that the lack of their product/failure to vote for their person/etcetera will RUIN YOUR LIFE FOREVER and lead to THE END OF THE WORLD!!! Subversions, aversions, and/or deconstructions are much less common and far more interesting, so please post only them or the most extreme examples of this trope.
On Monsters, Inc., after the first sighting of Boo, a professorial looking monster appears on TV, seemingly to calm people down. He then tells the audience "It is my professional opinion that now is the time to PANIC!!"
Aladdin: During the escape from the Cave of Wonders, Aladin tries to tell Abu that This Is no Time to Panic ... then sees the next obstacle up ahead and moans, "Start panicking."
Kung Fu Panda: As Tai Lung is effortlessly destroying every barrier that Vachir and his men made to prevent him from escaping, Zeng continually asks to run away, while Vachir just laughs at his cowardice. After Tai Lung gets past the last ditch defense...
Zeng: Can we run now?
Vachir: [Sobs in terror] Yes.
Films — Live-Action
Fatal Contact: Bird Flu in America was a Made-for-TV Movie about Bird Flu coming to America, while, at that exact same time, suddenly mutating into a form more easily communicable to and between humans. On the inverse side, very few people realize that Swine Flu actually is a pandemic already, because the definition of pandemic doesn't actually include death toll in the criteria, just how much it's spread and how hard it is to stop it from spreading further. The fact that it's spread just about everywhere but proven no more lethal than the regular flu is probably why a similar movie doesn't exist for it.
Not that the media didn't try, but it's hard to get scared of something when the vast majority of the deaths are innocent pigs.
Reefer Madness: in the film musical the Lecturer says "When danger's near exploit their fear." The one parent who objects to these Red Scare tactics is labeled un-American.
In Michael Moore's Fahrenheit 9/11, Moore replays a female scream after images of destruction to emphasize how the government exploited the 9/11 attacks to manipulate people.
Pretty much the whole point of The Host was satirizing this. Giant tadpole running around Seoul, eating people? It must be carrying A HIDEOUS NEW DISEASE! Let us completely focus on this possibility, and ignore the fact that it's eating people!
Played for Laughs in The Music Man. Salesman "Professor" Harold Hill's standard procedure is to work up a town into a frenzy by turning some trivial incident into a full-blown moral panic, so he can sell them on the idea of starting a children's band (and buy instruments from him).
Oh yes, we got trouble! Right here in River City! With a capital T and that rhymes with P and that stands for POOL!
Rollover: the public panics and riots after banks fail worldwide. 
Night Watch 1995: after the Hong Kong stock market computers are destroyed by a virus, economic chaos spreads across the globe. The stockbroker that is interviewed says: "Once Joe Average decides the market's unstable, they want out, at any price. They were telling me sell, sell, sell; but nobody's buying." Another report says the market has fallen 3000 points with no end in sight. 
In Airplane II: The Sequel, when Elaine announces to the passengers of the shuttle that they're out of coffee and a riot breaks out, the sign changes from "Don't Panic" to "OK, Panic."
Inverted in The Hitchhiker's Guide to the Galaxy, as the eponymous Guide has "Don't Panic" on the front in "large, friendly letters". But played straight in the fifth book, Mostly Harmless, as the latest version of the guide has PANIC in "small, alarming letters".
In Robert Asprin's humorous fantasy novel Myth Conception, as more and more horrible details come pouring in concerning the supposedly cushy job they've finagled, Aahz finally breaks down and invokes this trope with his apprentice Skeeve.
Live Action TV
Conspiracy Theory with Jesse Ventura: Every episode is about some threatening conspiracy with nefarious plans that always involve massive harm to the human population. The show will run clips of disasters while the narrator informs us that we'll die any day now thanks to yet another conspiracy.
The Leverage team once took down a TV reporter who made a career out of this. She actually turned down a fake story they fed her about secret terrorist prisons in the US because she didn't think it would scare her viewers enough. 
The X-Files, "War of the Coprophages": Scully attempts to control the mob with a speech about how they are giving in to panic. The mob ignores her and creates havoc.
Mock the Week also ran a story about Bird Flu, showing the scare-mongering of the English media.
Kryten from Red Dwarf has a "panic mode" for when the situation turns out to be more serious than he thought it would be.
The Animal Planet series Infested makes it look like infestations are caused by unstoppable pest species that will overwhelm the people dealing with said pests. In reality, such infestations are rare and most infestations can be treated rather efficiently.
A couple of episodes have focused on bats. Said episodes went out of their way to remind people "Bats have rabies! Your children could be in danger!". While rabid bats are very dangerous, you're more likely to be bitten by a rabid dog due to bats being nocturnal and encounters with them being quite rare.
Roseanne episode, Toto, We're Not in Kansas Anymore, when Lanford is under a threat of a tornado, Becky tells Roseanne that there was no time for jokes… promptly cue this sarcastic response by the titular character:
Max: Don't panic, 99. Whatever you do, don't panic! [Max calmly assesses the situation] Okay, 99, you can do it.
99: Do what, Max?
The entire point of Eminem's song "Without Me" is to parody this trope. Within the song he makes a Take That at the FCC, teases paranoid parents that he's "infecting in your kids' ears", and invokes The New Rock & Roll trope by alluding to the fact that the Moral Guardians also reacted with You Can Panic Now when Elvis premiered.
Hawkwind's production of Michael Moorcock's Sonic Attack, in which a robotic and oddly accented voice describes the lethal and agonising symptoms of sonic attack so you can be better informed about one happening in your neighbourhood. This is punctuated with authoritarian commands of "Do Not Panic!" repeated at intervals, as the voices become even more alien and robotic and the accompanying white noise grows louder....
Internet e-mail forwards and postings will often feature things from almost everything mentioned in the Real Life section and more. Examples are too numerous to list here (much like under Advertising) but can be found at such sites as Snopes and TruthOrFiction.com.
Parodied in a Calvin and Hobbes strip where Calvin's dad was watching "Eyewitness Action News", which aimed to tell its audience ''why YOU should be paralyzed with helpless fear!"
In the ending for the Portal 2 DLC, GLaDOS, upon seeing that the invader is in fact the bird, starts panicking and tells her robots to forget their training and flee.
In Crash Twinsanity, after removing Dr. Neo Cortex's brain and using it as a ball, Victor said that "This is the part where you run away screaming." Also cue Moritz doing a "go away" gesture to Cortex, before he did just so.
Simmons: Sir, the world is shaking apart, the ground is opening up to swallow us, and we're holding a bomb that will wipe us all out before any of that happens. I submit to you that if there is a time to panic, this is that time.
In the final episode of ReBoot - after Megabyte's return, Mike the TV urges everyone in Mainframe to "panic as only you know how to do!" It turns out this is actually Megabyte in disguise - all part of his plan to trap the guardians.
Sponge Bob Square Pants. Along with the page quote from "The Smoking Peanut", there's "Wormy" where SpongeBob and Patrick stir Bikini Bottom into a frenzy when they start warning people about a monster (actually a butterfly).
Patrick: Attention, Bikini Bottom! There's a big, scary monster that's going to eat you!
Also in the episode "Krab Borg", after Squidward learns that robots have "taken over the navy",
Squidward: Attention everyone, run for your lives! Robots have taken over the world! Fishes blink. Squidward: OUR WORLD!! crowd proceeds to panic
One episode features a hidden media conspiracy to actually release a disease and do reports on it blaming the imminent outbreak on cats.
In "Homer the Vigilante", Kent Brockman's initial report on the string of cat burglaries ends with him asking a professor if the town should panic.
Kent Brockman: Hordes of panicky people seem to be evacuating the town for some unknown reason. Professor, without knowing precisely what the danger is, would you say it's time for our viewers to crack each other's heads open and feast on the goo inside?
Vigilante justice, as vigilantes are almost always acting out of this when it's not direct revenge.
24-hour news networks in general. Specifically a lot of coverage of The War on Terror and terrorism post-9/11 has been accused of this; from some reporting, you'd think that there was a terrorist waiting behind every bush planning to assassinate you personally on your way to work.
Any "tabloid-style" television news, such as that popularized by Rick Sanchez at Miami station WSVN.
The coverage of the Great Recession, at least early on. You'd think from watching some of the reports that the whole world would implode at any moment.
In February 2009, Hugo Lindgren-writing for the New York Times-coined the term "pessimism porn" to describe these relentless predictions of financial doomsday and the public's fascination with them.
The absolute worst may come from a guy named Porter Stansberry, a supposed financial analyst who has been disciplined by the SEC for fraud. He's spent the past year flooding the internet, TV and radio waves with ads suggesting the that one of the most important dates in US history is coming and directing people to a website (whose URL is constantly changing). The site features a video with text and voiceovers only, explaining that the US dollar and economy will definitely collapse within the next year because all other nations are on the verge of agreeing to refuse US dollars (even though doing so would voluntarily ravage those countries' economies) and that the US government will respond by taking away everyone's money and sealing off all entry/exit to the country, so you need to get your money out of the country NOW (but then how will you get to it if the government is going to shut off access to other countries?). What's more, the narrator keeps saying that you MUST believe him because he correctly predicted the collapse of General Motors, Freddie Mac and Fannie Mae (news flash - a lot of people did). Oh, and did we tell you he knows the one thing that will not only save you but actually make money when this occurs, and he'll send you the booklet about it for free - IF you sign up for his $50 a month newsletter? (Spoiler alert - it's "buy a farm.")
Every Environmental Scare Ever. According to Paul Ehrlich we've all died of skin cancer from UV exposure thanks to aerosol cans, the world is so crowded people are falling off,and there's no clean water left to drink— oh, and the ever popular 'we're running out of resources!'. These issues are real, but irrational attempts to solve them can and have made matters worse.
Every few months or so, an article comes out saying that Facebook, Twitter and texting make children and teens less socially interactive/have less empathy/have poorer grammar (despite the fact that you could arguably say that about anything that allows you to communicate without actually talking to the person). This may or may not be true, however the way the media goes on about it you would think all people under the age of 20 who use social media are anti-social basement dwellers who spend all their time "sexting". This is probably a continuation of the fear when MySpace was popular that all people children meet on there are pedophile serial killers.
Every time a new doll aimed at preteen girls comes out, someone will freak out and claim that the dolls are promoting loose morals and/or flawed beauty standards.
A guy came on Dr. Oz and speculated that apple juice was harmful to drink, mostly because of arsenic being found in the stuff. The guy who made the claims never mentioned there are TWO common compounds of arsenic, one harmful to the body, and the other doing relatively nothing.
Every couple of months, there will be an article in some magazine or a book by some non-writer about why African-American women can't seem to get married. The reasons cited will usually be the same: 1) All the Good Men Are Gay (or in prison) 2) Black men all marry white women 3) Black women are too picky 4) Black women are just plain un-marriageable (or ugly, in extreme cases). The solutions will always be the same: 1) BW should only date outside of their race or 2) BW should settle for the high school dropout on parole with 11 baby mamas. Essence magazine in particular gets a lot of heat for this.
There are a LOT of magazine articles and books that exploit the fear that black women have of black men dating/ marrying women that are anything other than black. Almost to a predatory extent. Like the idea that black men go to Brazil not to experience the culture, but to hook up with Ambiguously Brown women that look like Nicole Scherzinger.
On a more general related theme. Periodically, upmarket British Newspapers such as the Guardian and the Observer run angst-laden fearful articles about why otherwise hyper-successful professional women who otherwise have got it all are in a state where, unaccountably, they are still single. These articles are invariably written by self-styled hyper-professional successful women who otherwise have got everything going for them, ie by newspaper journalists and opinion columnists who unaccountably remain single despite being young(ish), well-rewarded, attractive, and successful. They invariably come to a gloomy conclusion that all the best men are taken or else gay. Leaving only hopeless losers and deadbeats with inflated opinions of themselves. Of course, as we are successful in every conceivable way and attractive with it, it cannot be our fault that we remain single, despite being pestered by all those horrendous losers and ugly guys who just haven't got the message yet. There just aren't any suitable men out there...
Disasters that could wipe out the world or at least humanity get a huge amount of coverage from edutainment networks. Most sensible people agree that it's going to happen someday. TV experts keep stressing that 'someday' includes tomorrow.
There is always some "teen craze" that is being hyped by the news media. Rainbow parties, bracelets telling what sex acts you've done, vodka gummy bears, vodka tampons, etc. Almost all of them are not as common as the news media would have you believe. If anything the news just makes teens want to try it.
In the case of Rainbow parties, no one actually did them. If there had been a few cases, then it was because they got the idea when the news claimed that everyone was doing it (a claim that was based off of one story in a book).
The whole infamous fiasco involving The West Memphis Three is one particularly disgusting example of how the media and rumors can cause innocent people to go to jail. During the time after the murders the media and people throughout the neighborhood spread claims about how "the murders were part of a satanic ritual!!!" and "those three teenage boys are into rock and roll and bad emo stuff like that, they must be SATANISTS!!!". By the time of the conviction pretty much everyone was so worked up over all the satanist conspiracy crap that the conviction of the boys was a slam dunk even though the best evidence was that of the boys confessed (and later discoveries suggest that confession may have been coerced).
The mass panic over the U.N. Small Arms Trade treaty suggested that U.N. troops or the Department of Homeland Security would personally come to your house and take your guns away, despite the U.N. not having any real power, America being a superpower that can ignore the U.N. if it wants, U.S. law specifically stating that our Constitution overrules any treaties, 'the Treaty itself explicitly saying (in effect) "None of this infringes on gun rights that may or may not be enshrined in any country's constitution or laws," and the Treaty not even being about private firearm ownership, but rather international arms trade: The provisions amount to "Every country has to have records regulations. This to ensure that gun manufacturers have to make sure that whoever's buying their guns won't sell them to rebels, rogue states, organized crime, and other international bad guys." As a happy bonus for American gun retailers, sales ironically went up, as they often do whenever there is a threat (real or otherwise) to 2nd Amendment rights.
It is joked that a Democrat in the White House is the gun seller's best friend, as his existence ensures heightened demand for guns.
Ever since SNOWPOCALYPSE (the moniker given to the unusually-harsh winter in the US in 2010), The Weather Channel has been naming winter storms like hurricanes and giving them almost as much weight in their reports. It's no longer enough to just say, "Heavy snow will be coming through over the next couple days." Now it's, "WINTER STORM GANDOLF!!" They seem to have pop culture and mythological naming conventions (Khan, Nemo, Athena, etc.), which only makes it that much harder to take seriously. The government-run National Weather Service (The Weather Channel is privately-owned) pointedly refuses to use those names.
However, the NWS has gotten criticism with earlier and earlier naming of Tropical Storms, with some being named even in the Tropical Depression stage. This has lead to accusations of sensationalism.
Homophobia and transphobia carry shades of this when they are used as "culture war" issues. Specifically, the All Gays Are Pedophiles canard or "guys in dresses in girls bathrooms" being paired with Pædo Hunt are very nasty variants, or All Gays Are Promiscuous being used to start a moral panic about promiscuity (never mind that heterosexuals can be just as promiscuous).
During the 2012 election, many believed that the Republicans wanted to ban tampons, based on a satirical article on the Freewood Post.
Millennials have become a popular media target for inducing panic, especially when it comes to discussions about them in the workplace. From the way the media talks, you'd think that anybody under the age of 35 is a self-centered, basement dwelling text-aholic. In reality, such stereotypes (at least, to the extremes by which the media depicts them) apply to a very small number of millennials.