STOCKHOLM—Spotify AB is expanding its streaming-music service in Asia, extending its push for a more solid global footing than its rivals as the company continues to post losses.

The Stockholm-based company, one of the largest streaming-music providers in the world, on Tuesday began offering its service in Taiwan, its fourth Asian market, after Hong Kong, Malaysia and Singapore. Spotify also rolled out its service in Turkey, Greece and Argentina. The company has more than 24 million active users, six million paying subscribers...