The dispenser for a fluid product such as a powder from single dose holders, e.g. in the form of blisters in a strip (21) or disc, has a holder opener (40) in the form of a piercing/cutting mechanism with a mobile support (50) that presses each holder against the mechanism as it is moved in a curved trajectory between non-dispensing and dispensing positions by a spring (51) or flexible strip that is released by the user prior to inhaling.