Line dancers move across the floor of the Eagles’ Nest Rockin’ Country Bar in a series of grapevines, jazz boxes and shuffles, stepping to the beat of “Good Time” by country musician Alan Jackson.

Performances in N.Y.C. It’s happening in bars, dance studios, even a roller rink. And you’re as likely to find the crowd clad in cutoff shorts and crop tops as boots and denim. It’s happening in bars, ...