I think a fair amount of anime has one of the voice actresses in the show sing the title tune, or the "Insert songs". The above Ranma clip for example.
Bubblegum Crisis for example, and the main character sings as part of a band as well, although BGC doesn't have a fixed theme song or opening until the second OAV series and later the TV series.
One of the later City Hunter episodes has Ryo singing the first title tune while walking home.