A "singing career" refers to a person's profession or line of work as a singer. It means that the person earns a living or makes a career out of singing professionally. Full definition
And yes, Heigl sings a bit too... but not much considering her character supposedly had a successful singing career in years gone by. (moviereviewsfromthedark.com)
Gomez is known for starring in Disney's «Wizards of Waverly Place» and movies like Netflix's «The Fundamentals of Caring» and leading a successful singing career as both a solo artist and frontwoman of Selena Gomez and The Scene. (rismedia.com)
Karina has written and directed films, taken on prominent stage roles (including in plays directed by Jacques Rivette and Ingmar Bergman), had a successful singing career, and written four novels, but, of course, she is cited most often as Jean - Luc Godard's New Wave muse and first wife. (criterion.com)