The film that features the song “New Year's Eve” and follows several New Year's stories in New York City is New Year's Eve.
Directed by Garry Marshall, the 2011 romantic comedy is an ensemble film whose interconnected stories unfold during New Year's Eve in Manhattan. Its cast includes Halle Berry, Jessica Biel, Jon Bon Jovi, Ashton Kutcher, Sarah Jessica Parker, and Robert De Niro.
The film uses recognizable New York locations and traditions, including the Times Square Ball Drop. Its characters deal with romance, family relationships, career decisions, grief, and second chances as midnight approaches. The structure is similar to Marshall's earlier holiday ensemble film Valentine's Day, but the setting and date are different.
The title can create confusion because it is also the ordinary name of the holiday. The specific answer here is the 2011 film, not a general description of a New Year's Eve event or another movie with a similar holiday theme.