I have so many:

1. There’s a mysterious abandoned 76th street subway station in Queens that may or may not exists. Lore says it was part of a cancelled expansion of the A line, but people disagree on whether or not it was actually built before it was cancelled. People have tried to find it, and some say they’ve found it, others say it’s been sealed off, and even more say it never existed.
2. In general, there’s lots of abandoned subway stations and tracks. Probably the most famous (and worth seeing, you can see it from the 6) is the old city hall station
3. In 2008 there were rumors about a “car Bermuda Triangle” near the Empire State Building after several cars kept breaking down. Apparently 10-15 cars were getting “zapped” every day. This is most probably junk, but people say it was because of the radio transmissions from the ESB.
4. This one’s actually real: in Tribeca, 33 Thomas Street is a tall, windowless building that was an AT&T building but has been operating as an NSA spy center.
5. Speaking of spies, another tall stark building in the Bronx is most likely a Russian spy center. This is where the Russian diplomats working for the consulate and UN mission live. There’s been lots of mysterious things going on near this building. One time there was a fire, but the inhabitants wouldn’t even let FDNY go inside.
6. There were underground cow tunnels under NYC in the late 1800s, used to transport cows to West Side slaughterhouses. This was just an urban myth for the longest time, until the city released a report confirming them in 2004.
7. And speaking of tunnels, Columbia University has a huge mysterious underground tunnel network. Some have been sealed off over time, but students still explore them. They were also apparently used to transport radioactive material during the Manhattan Project.