Basically, City Noir + Big Applesauce = The Big Rotten Apple.
New York City is depicted as a dysfunctional, crime-plagued, vermin-infested, smog-choked, polluted, grimy, sleazy, seedy, corrupt, racially-divided, poverty-ridden, morally-and-financially-bankrupt Wretched Hive filled with Apathetic Citizens, hostile Jerkasses, violent psychotics, drug addicts, deviants, a crumbling infrastructure, and not enough parking spots.
And the worst thing is, once you've gotten a taste of it, you can't stand to live anywhere else.
The Big Rotten Apple trope can come into play in any story set during the city's existence but you'll most often see it in stories set in New York in the years before the economy began to pick up steam and the nation's cities started to recover (roughly the period lasting from the mid 1960s to the early 1990s). You'll also sometimes see it in stories set in late 19th Century/Gilded Age New York.
See also Brooklyn Rage, and the Place Worse Than Death section for New York.
Anime and Manga
Mad Bull 34 portrays the Big Apple as a crime-ridden hellhole, where the only difference between the criminals and the police are uniforms.
In Daredevil, the Hell's Kitchen neighborhood of New York has this element.
The Punisher: The change in the city since the 70s and 80s is acknowledged at one point during the "Welcome Back, Frank" story arc when Frank, during an inner monologue, reflects a bit on how the city's cleaned up quite a bit on the surface, but at the core it's still as rotten as ever.
Gotham City is an expy of the darker side of NYC in the Batman franchise. ("Gotham" has been a well-known nickname for New York City long before Batman even came about, and artist Frank Miller has referred to Gotham City as "New York City after dark.")
Watchmen: Rorschach gives his opinion on the city:
"The streets are extended gutters and the gutters are full of blood...Beneath me, this awful city, it screams like an abattoir full of retarded children. And the night reeks of fornication and bad consciences."
NYX is set in a particularly nasty, crime- and gang violence-riddled part of New York City.
The fanfic series Ultimate Sleepwalker: The New Dreams and Ultimate Spider-Woman: Change With The Light are both set in the same alternate version of the Marvel Universe. This universe's version of New York has both the highest number of supervillains per capita and the overall highest crime rate in the entire United States. It's specifically noted that Rudy Giuliani's law enforcement initiatives were a dismal failure, and the city remains as much of a crime-ridden hellhole as it's ever been.
The Out Of Towners(1970): An Ohio couple journeys to New York and finds a rundown metropolis beset by crime, ineffectual police, strikes, institutional incompetence, and a callous populace.
Serpico: Especially as the protagonist of the film and the non-fiction book it's based on keeps getting told that corruption in the NYPD is a matter of a few "rotten apples" rather than "the barrel itself being rotten."
Death Wish and its New York-set sequels depict a city where violent crime is so out of control that citizens are forced to take vigilante action.
Taxi Driver: Travis Bickle, a night taxi driver, sees the mid-1970s city as this.
All the animals come out at night - whores, skunk pussies, buggers, queens, fairies, dopers, junkies, sick, venal. Someday a real rain will come and wash all this scum off the streets.
The Warriors portrays New York as having an entire alternate society of warring tribes (which includes the police) that only come out at night. There is also a game adaptation showing how the cops are largely complicit in underworld dealings.
Fort Apache The Bronx is based on the real-life nickname of the 41st Precinct (South Bronx) of the NYPD in the late 70's and early 80's.
Do the Right Thing depicts a late 80s Brooklyn neighborhood as a racially charged tinderbox ready to explode.
Joes Apartment. The eponymous hero gets mugged three times in a row. ..Before leaving the bus station.
This trope can also be seen in films that take place during the 19th century that show the gritty side of 'Gilded Age' New York:
Gangs of New York mostly takes place in the historic Five Points slum, which is filthy, violent and crime-plagued, ruled over by rival gangs and a corrupt police force. And then the Civil War Draft Riots break out and the area gets destroyed by cannon fire (which really happened).
An American Tail has a New York ruled over by cat gangs (representing racial persecution in historic New York at the time) extorting immigrant mice with a protection racket. The sequel accentuates New York's negative characteristics to force the Mousekewitz family to move out west.
Teenage Mutant Ninja Turtles depicts New York in the middle of a huge crime wave perpetrated by Shredder's Foot Clan, made up of teenage runaways and orphans. The sequels don't have this trope so much though.
Batman: Anton Furst's Gotham City is New York "if Hell erupted through the pavements and kept on going." It's also inspired by, in Furst's words, "an immigrant's vision of New York City."
Ghostbusters, its sequel, and the Video Game. Diners at Tavern on the Green ignore a panicked Louis Tully as a Terror Dog possesses him; the combined negativity of the city's denizens manifests as a river of slime; the city is routinely attacked by ghosts of its own homeless and construction-workers...the list goes on and on. Worst of all is the feeling of a city without spiritual direction, and the city's only contact with the other side is mischievous at best and outright malevolent at worst. Subverted in the sequel's climax, when the Ghostbusters use the Statue of Liberty to awaken the good feelings lurking beneath New York's crusty exterior and combat the slime's negative influence.
In Coming to America, this is possibly invoked. Prince Akeem and his best friend Semmi stay in New York in order to meet an American woman. Akeem insist on staying in a dirty apartment in a rough neighborhood in Queens so people won't realize that he is royalty.
"Not since Jason Takes Manhattan have I seen a film that was seemingly designed to scare people from wanting to go to the Big Apple (oddly, the trailer inexplicably points out that NY has the lowest crime rate of the nation's ten biggest cities). During their first date, our leads go into a convenience store (pretty much the first time we see them leave the school) and instantly the place is held up, with the robber and clerk ultimately shooting each other. Then she is mocked for being upset, as if a NY resident should be used to people being shot to death right in front of them whenever they walk out the door. All clerks that they encounter are jaded assholes, and every place they visit looks like a hellhole. I mean, the scavenger hunt was actually designed as a real one for new students - shouldn't he have chosen landmarks for them to visit in order to find clues instead of closed down dance studios and anonymous seedy bars? And even odder, when one girl is killed, her body is left in plain sight on the sidewalk for a good chunk of time, undisturbed, which just suggests that a corpse is nothing unusual in the city."
Made fun of in the beginning of Mostly Harmless, which proceeds to outline all the ways in which New York is a terrible place to live if you care about your quality of life.
In Animorphs The Familiar, Jake wakes up in a dystopian, Crapsack World version of New York where all of humanity has been enslaved by the Yeerks and the only free people are an underground group of rebels.
Lyndsey Faye's novel The Gods of Gotham, is set in mid-nineteenth century Manhattan like Gangs of New York and follows one of New York's first cops whose beat is in the squalor and poverty of Five Points.
Hubert Selby Jr.'s Last Exit to Brooklyn portrays the titular borough in the late 1950s as a Wretched Hive of prostitutes, drag queens and criminals. It was quite shocking when first published in the early 1960's.
Barney Miller takes place during the 70's and early 80's. The bureaucracy, high crime, and perpetual budget crisis make for great comedy and occasional drama.
Late Night with David Letterman and Late Show: Before September 11, 2001, the opening would have the announcer make some disparaging crack at about the city ("From New York, where the subway cars smell like urine, it's The Late Show with David Letterman!"). Post-September 11th though, every episode begins, "From New York, the Greatest City in the World...."
Saturday Night Live: Since the show airs from New York, it has often invoked this trope during much of its run.
For a short time during the early 80s, the opening credits announced, "From New York, the most dangerous city in America, it's Saturday Night Live!"
A sketch from the 1990-91 season featured guest-host Joe Mantegna as an oblivious city official being interviewed on a radio talk show who deflected caller complaints about the city's increasing crime rate (including one from a caller who was mugged and assaulted as he was waiting on the phone) by pointing out where else could they see great Broadway shows like Cats.
The BBC America series Copper is set in the same time and place as the film Gangs of New York (a few months after the Civil War Draft Riots), so it deals with the same rotten apple.
Detective show Cagney & Lacey derived much of their drama from being in the Rotten Apple in the late 70s and early 80s and the various conflicts, gang turf wars, and poverty related crimes formed the bulk of the station's workload.
A background theme of Mad Men is the decline of New York City during the 1960s. In season 2 (set in 1962), 22 year old Peggy's biggest concern about riding the subway alone at night was that the bamboo seats would ruin her stockings. In season 6 (set in 1968), sirens play in the background of almost every scene, the Drapers' Upper East Side penthouse is robbed, and Peggy's boyfriend is stabbed while waiting at a bus station near their home on the Upper West Side.
Lampshaded in a season 4 (set in 1965) episode by Joan and Roger. They go to a diner that they used to frequent back in the 1950s (also on the Upper West Side) and note how much seedier looking its gotten since the last time they went. Of course, that dinner ended with Joan and Roger getting mugged.
Billy Joel's song "Miami 2017 (Seen the Lights Go Out on Broadway)" is a partially-satirical vision of what might happen if America just plain gave up on New York City and ordered it to be completely demolished. The tone shifts throughout, but the second verse mockingly suggests that some places, namely 42nd Street and Harlem, might be none the worse off for it.
Word of Godis that the album cover of Billy's 52nd Street was photographed in a setting (the dingy, weatherbeaten walls of a New York City jazz nightclub) to reflect both the jazzier sound of the music (Billy's holding a trumpet, like a jazz musician), and the time period it was recorded in (late 1970s, around the time of "Son Of Sam").
Also to some extent "Doo Doo Doo Doo Doo Doo (Heartbreaker)".
Walk on the Wild Side and Dirty Blvd. by Lou Reed.
The whole New York album, really.
New York's Not My Home by Jim Croce describes New York as a wretched, depressing place.
Also, Box no. 10 (Croce obviously wasn't overly enamored with the city).
The Message by Grandmaster Flash & the Furious Five, which was about the pressures of living in the inner city. It's taken further with New York, New York, which explicitly names New York and all the crime and poverty: having to eat dog food to stay alive, homelessness, criminal insanity, single motherhood and abandonment, stick-ups, prostitution...
Too much. Too many people...
Let's Go All The Way by Sly Fox:
Living in New York
Looks like an apple core
The video for New Order's "Confusion" shows New York as it existed in 1983, including graffiti-strewn subway cars and a Times Square full of porno theaters.
The Lonely Island song "I Run NY" mocks this; it is sung from the perspective of New York's mayor and his complaints get progressively more mundane as the song progresses (at one point he nearly cries because the Chief of Police mocked his tie).
In a column he did in retaliation for a New York Times piece on Miami being a crime-ridden druglord paradise, Dave Barry tells of researchers from New York asking people why people didn't like New Yorkers and being told everyone was so rude. "Then the researchers spat on them."
The small press RPG Fates Worse Than Death posits a "street level cyberpunk" future where Manhattan has gone bankrupt. Anyone who lives in the decayed, dangerous island either commutes in from heavily-secured suburban communities, is a natural thrill-seeker, or is just too damn poor to move out.
Both the theatrical and film versions of Jules Feiffer's pitch-black satire, Little Murders, feature an over-the-top depiction of New York as a decaying urban hell-hole beset by continual garbage strikes, electrical outages, and numerous unsolved random murders.
Although the series as a whole is commonly known for depicting fictional counterparts of New York City, Liberty City, as vice cities, the Grand Theft Auto III version of the city evokes this trope the strongest by portraying itself as "The Worst City in the World" with heavy pollution, rampant crime fueled by a menacing new drug, widespread corruption, increasingly militant tramps, and no tourist industry to speak of. Grand Theft Auto Advance even portrays the city as having been hit by bubonic plague. In the Turn of the Millennium.
Grand Theft Auto IV toned it down somewhat, given that it's based more on the "cleaned-up" New York of Rudy Giuliani and Mike Bloomberg rather than the Wretched Hive that it was in The Seventies and The Eighties. Of course, this means that this trope has been replaced with stereotypes of modern New York, such as the city's gentrification into The Theme Park Version of itself, its "nanny state" attitude to things like guns and junk food, and its post-September 11th police presence. Case in point: the Statue of Liberty's stand-in is a monument to "Happiness", i.e. crass commercialism.
Max Payne takes place in an Anachronism Stew of seventies-era Manhattan — when the city was a throne to vice and grunge — and the slick corporatism of modern NYC. The mob is so powerful that it controls the police and the press, and Aesir Pharmaceuticals controls the mob.
The Nostalgia Chick seems to agree that NYC is rotten but loves the city anyway, as movies that depict it as clean always draw her ire, and the insults in "Fairytale Of New York" make her nostalgic. (She was in Los Angeles at that point.)
Thanks to the worse economy, New York City in Fear, Loathing and Gumbo on the Campaign Trail '72 goes bankrupt in 1975 and has its finances taken over by the federal government, on top of all the other problems that it faced in The Seventies. Things are no better The Eighties, where it gets taken over by Mayor Spiro Agnew, of all people, who institutes a policy of walling off the worst ghettoes and arming the NYPD with belt-fed machine guns and other high-powered military hardware.
The Simpsons: In "The City of New York vs. Homer Simpson" episode, Homer tells about a previous visit to New York during the 70s when this trope was in full force. A pickpocket stole his wallet, a police officer stole his suitcase, a pigeon stole his hot dog, Woody Allen dumped garbage on his head, and he got chased through the streets by an angry pimp until he fell in a sewer.
Homer: ...and that's when the C.H.U.D.s came at me.
Marge: Of course you'll have a bad impression of New York if you only focus on the pimps and the C.H.U.D.s.