First seen in Star Trek V: The Final Frontier. And yes, it is totally swooshable.
About this creation
In the late 2280s, the Enterprise-A carried four of these standard shuttlecraft. We see two of them on screen: the Galileo 5, and the Copernicus 3.
A hatch on each side allows quick ingress/egress of as many as 10 people, with bench seating in the back.
I have yet to make the era-specific uniform decals, so TOS Scotty is stepping in to help out.
An aft ramp provides additional (and totally redundant) access.
The full-size mockup of the craft was altered after filming to become the Type 6 shuttle on Star Trek: The Next Generation. The centre section with the side hatches was removed, and the warp nacelles were changed. Some new livery and the addition of side windows and voila! A shuttlecraft from 70 years later.
The filming model was also altered, and appeared only once in a cut scene from Star Trek: Generations (the movie), this time as the Type 5. It retained its full shape and the original nacelles, but with updated livery and lighting effects. It too had the additional side windows. But, since it was cut from the final film, it's not really canon.
UPDATE: I finally made some era-appropriate uniform stickers.