Surprisingly, a lot of modern major Taiwanese singers/bands have some songs in Taiwanese. They differ fundamentally from Cantonese songs: Taiwanese songs are mostly written using oral Taiwanese, whereas Cantonese ones are mostly written using standard Chinese.
Here are three modern Taiwanese songs which I recommend, by Mayday, Jay Chou and Sodagreen:
FYI, the one by Jay Chao is partly in Mandarin; the other two have separate Mandarin versions, presumably to appeal to the non-Taiwanese speaking masses. Enjoy.