This is a hybrid approach...a mouth I designed for a puppet a couple years ago (it's actually a prototype, so it's a little rough around the edges).
The upper mouth plate is rigid and made from thin, but stiff plastic. There is also a narrow rigid piece at the edge of the lower mouth plate. The two are joined together by a piece of flexible material (some kind of vinyl I think) and it was very sturdy, but also very expressive.