Mythologies of New York City
As at least a temporary resident of New York City, it seems important to find traces of cultures, legends, and mythologies which make the city more dynamic for my wonder-starved mind. I am someone who likes tales, stories, what-ifs, legends, forgotten histories and imaginings, so finding them in new homes brings me closer to those places.
New York is a city packed with such histories, but they are hard to find, because the city of hustle is always deleting their breadcrumb trails. But they can be found.
Like so much of New York's cultural history, this name was invented by Washington Irving. The name was originally meant to lampoon the city and its denizens, as "Gotham" was supposedly a term used by the English to refer to a city filled with fools.  Irving may or may not have also realized that the idiocy of the fictional, English Gotham was of a somewhat ruse to trick the King of England.
Today, Gotham is best known as the setting of the Batman universe, but of course the city of Batman is itself a certain, grim take on New York. Sites like the Gothamist show the term in continued practice. 
The term "knickerbocker," now best known as the name of Manhattan's pro basketball team, originated as an early pseudonym of Washington Irving. The full name was "Diedrich Knickerbocker" and was used to name a character- supposedly a lost Dutch historian who went missing in Manhattan around 1809 - that Irving implied wrote a book called "A History of New-York from the Beginning of the World to the End of the Dutch Dynasty" available here 
Irving distributed missing person flyers around the city to raise awareness of the character and his disappearance. He also forged a letter from the hotel owner where Diedrich had been staying, claiming that the owner would publish the Dutchmen's manuscript if he failed to pay his bill.
Irving subsequently published the text to great popularity. It was a satire of NYC politics.
In 1946, having become a common demonym, the city's basketball team was named the "knicks." 
Lost Subway Lines & Stations
With over 840 miles of track, the MTA operates the largest subway in America through the boroughs of New York City. It's ubiquity and familiarity make it an indispensable site of shared cultural experience. Familiarity with the system is a sign of belonging, and preferences for certains lines operates as a way to show regional allegiances.
When in doubt, New Yorkers small talk about the subway. AND, more importantly, about its future/past.
The lost subway lines and stations of the system are a major point of discussion and imagining. This has been recently revived by work on the Second Avenue line (the T) which is expected to change the fabric of the city by placing new commuting potential into the far east side .
WNYC recently broadcast a program about the forgotten lines and stations of the system, offering a fantastic interactive map . Several blogs followed up with thoughts about the scale and expensive of the early system versus that of today 
Curiously, looking over the comments of these posts led me to the fabled 76th Street Station which is apparently a highly divisive issue among the seekers of lost subway culture. In a 2003 New York Times article, the case is made for the existence of the station, despite no distinct evidence .
The sheer idea that a station COULD exist, but cannot be proven in either way shocks me as some much in the city is black and white. But the sheer vastness of rail line in the city, along with gaps in memory, history, and archives, creates the continued possibly for urban caverns to exist where no one even thinks to look.
Rats (Giant and Otherwise)
Wildlife perturbs New Yorkers because the city has been so urbanized and industrialized that any lingering symptoms of the natural world threaten developed order. This is particularly true of animals who have flourished within the city's hyper-urban space rather than succumbed to its pressures.
Rats are the most ubiquitous and common "worry" animal. No one fears squirrels, but a subway rat suggests disease and nonchalance. They have naturalized the unnaturalized terrain of New Yorker's subterranean and therefore subconscious life.
Rats per person
One popular New York subject is the "x rats per person" stat. This will be quoted between 10 and 15, in my experience, and research suggests Health officials know it is at least 1 per person .
Talking up the size of rats is also popular. And it can be front page news when someone FINDS a huge rat. Like this one found in the Bronx  who was probably the brother of this one found last fall .
Giant rats are often confused with "Rat Kings" which are actually giant masses of rats entangled by their tales in a terrifying macro-organism. This is also highly speculative, but it's so popular a concetp that it shows up in a 30 Rock episode .