Mandarin is most widely spoken, but outside of Taipei it is common to hear Taiwanese dialect (it's about as different from Mandarin as Cantonese). Older people usually know it and use it often; many younger people know very little.

The Taipei subway makes announcements in 3 languages, Mandarin, Taiwanese, and English. I have heard that CI and BR do announcements in Taiwanese also but I can't confirm if or how often.