* Solo vs. Featured: Do we count only songs where she's the lead artist, or do we include collaborations where she's featured?
* Official Releases vs. Unreleased: Should we only count songs released on albums and singles, or include unreleased tracks or demos?
* Song vs. Version: Does each remix or live version count as a separate song?
That said, according to official sources and databases, Rihanna has released over 100 songs, including both solo and featured tracks. She has also recorded many unreleased songs.
To get a specific number, you would need to define what counts as a "song" for your purposes and then consult a database like Discogs or AllMusic.