The most appropriate Wiki links have been provided in the comments to the question above which should already provide context. I was going to let it go, therefore, until I saw this topic treated in FiveThirtyEight in this article.
They mapped out a potential State of New York City along the lines of Norman Mailer's 1969 New York mayoral platform:
Part of Norman Mailer’s 1969 New York mayoral platform was a petition for the city to secede and become the 51st state.
That's my reasoning for choosing this as the most serious push. The platform on which Mailer ran is further described here. Wikipedia further gives us these posters (this and this) from the campaign:
Mailer finished fourth in the 1969 Democratic primary, and not above fourth place in any of the five boroughs.