New York, New York, so good they named it twice - and sang about it a countless number of times. From Across 110th Street (Bobby Womack and Peace) to The Zoo (The Scorpions), the city has been an inspiration to a multitude of musicians, throughout the decades and in most genres.

U2 found an angel in Harlem, The Pogues saw a fairytale in the city's Irish history and Frank Sinatra wanted to be part of it and wake up in a city that never sleeps.

Broadway, Brooklyn, Central Park and Manhattan have all be immortalised in song many times over, and even Wall Street has been given a namecheck by the likes of 10cc and Yello. Thoroughfares from 6th Avenue through to 153rd Street (with many alighting on 42nd) have been lyrically trod time and again.

In 1976, when Billy Joel found himself in a New York State of Mind, he was returning to the city on a Greyhound bus, on the Hudson River Line. Unless things have changed a lot since the 70s, Joel wouldn't have had much choice if he wasn't prepared to fly; the Hudson is devoid of river traffic as I stroll along its Upper West Side bank.

Not a vessel moves on either the New York or New Jersey side as the cold winter wind whips up the river. Chesley Sullenberger, the airline pilot who became a hero when he landed his crippled airliner on the Hudson in January 2009 (and thus became himself the subject of a song: A Real Hero, by some outfit called College feat. Electric Youth), couldn't have wished for a more peaceful stretch of water to ditch in.

The riverside path is almost as empty as the water, the occasional jogger, cyclist and Canada goose making the most of the solitude. The effect is calming in a way a stroll along Tsim Sha Tsui's Avenue of Stars could never be.

Approaching Broadway, I stop for a coffee at the Olympic Flame Diner. New Yorkers are not short of confidence and so tend to speak loudly, providing dialogue for anyone people-watching in a public place. As I sip, a flamboyant black man in the booth across the aisle is trying to get a large white lady to be more trusting of her singing voice.

"Just sing," he says, in a booming voice, "just sing". And then he does, with operatic intensity. I appear to be the only one in the busy diner to have noticed.

It may not be hard nor far to reach, as the Ramones claimed, but it's too chilly to hitch a ride to Rockaway Beach, which, I'm reliably informed, has yet to fully recover from the battering it took during 2012's Hurricane Sandy anyhow. For a taste of out-of-season seaside glamour, I instead ride the F train to Coney Island (favoured by Herb Alpert and Death Cab for Cutie, among many others). Except the F train doesn't go all the way to Coney Island today, so I get out at the mysterious sounding Avenue X and walk the few blocks it takes to get to the shore. This isn't a mistake, as such, but the stroll is a reminder that what lays immediately behind urban seaside hot spots tends to be gritty.

The boardwalk is almost as deserted as the banks of the Hudson. The amusement park and most of the other attractions are shut for the winter, as is, says a sign, the sweeping beach, but there is a sale on at the Brooklyn Beach Shop and Tom's Coney Island is trying to entice the few hardy souls who have braved the chill with all-day breakfasts and, rather optimistically, waffle ice-cream sandwiches. I order some pancakes and sit down to work out which subway lines I need to take to get from this Tom's to another one, for a more substantial meal.

Tom's Restaurant, in Morningside Heights, was made famous as Tom's Diner by Suzanne Vega (and even more famous by television sitcom Seinfeld, in which it featured). The Mexican football team are playing live on the TV as I take a seat at the counter (singletons are not allowed to "do a Vega" and sit in a booth by the window). The staff are mainly Hispanic and a vocal, local crowd are taking an interest in the game alongside them - and an even keener interest in the chicas the cameramen pick out in the stadium seats. Tom's is a cheap diner, serving cheap diner food, but, that understood, its corn beef hash hits the spot. I take a quick look at the Seinfeld memorabilia behind the bar and venture out before the final whistle is blown in Mexico City.

The temperature is not much above zero, and that's at street level. On the 86th floor observation deck of the Empire State Building the air is bitter. But it's also remarkably clear - something we're unlikely to ever witness from The Peak, given Hong Kong's appalling pollution.

"In New York / concrete jungle where dreams are made, oh / There's nothing you can't do / Now you're in New York," sings Alicia Keys on the hit single she had with Jay-Z in 2009, Empire State of Mind, a regular topper of charts compiled of New Yorkers' favourite songs about their city.

That concrete jungle stretches away to the horizon from the Empire State Building, the city's own soundtrack of emergency-vehicle sirens muted a little by the height. To the north, ravines of tarmac dissect high-rise blocks up to and around Central Park, the whole of Manhattan bordered by the Hudson and the slightly busier East River. To the south, looking like a tiny green girl in the distance, is the Statue of Liberty, who, over the years, has welcomed scores of immigrants arriving with the hope that their dreams, too, would be made here.

Standing atop such a bold structure as the Empire State Building and surveying the whole of Manhattan all the way up to the Bronx and over to Brooklyn, even through wind-stung eyes, gives a sense of the vigour, self-assurance and, yes, near-limitless possibilities engendered by a city that still welcomes anyone wanting to seek their fortune - or write a song.

Also to the south is the building judged only the day before to be the tallest in the United States. The new World Trade Centre represents the resilience of its city. From the tumult of the civil war and the 1863 draft riots (part of the context behind the 2002 movie Gangs of New York) through the Wall Street crash of 1929 and the days when visitors had to take a walk on the wild side while The Velvet Underground waited for their man near the intersection of Lexington Avenue and 125th Street, to modern tragedies 9/11 and Hurricane Sandy, setbacks and hard times seem only to make the Big Apple stronger. Might it be that resilience - and the energy and confidence it gives rise to - that songwriters find so irresistible?

"These streets will make you feel brand new," sings Keys. "Big lights will inspire you.

"Let's hear it for New York."