Imagining contortions of the vertical/horizontal stabilizers and possible use cases for dodging or acrobatics, I was wondering if there were any known aircraft with such a design.
The SR-71's rudders pivoted without any fixed leading edge, as shown:
The T-38 (along with many other supersonic jets of its era) had a pivoting horizontal stabilizer with no fixed leading edge:
So yes, each of those features asked about can be found in production aircraft. It seems likely that some aircraft sometime even had both, although offhand I don't know which one.
Many aircraft use an all-moving horizontal tail—with or without an elevator.
Transport aircraft (B737, A320, etc.) often use an all-moving tail with an elevator. In this case, the tail moves very slowly on a jackscrew—it is moved for trim, with faster rate control still accomplished with the elevator.
There is also a concept called a 'free wing'. It mounts the wing on a bearing through the aerodynamic center. The wing is free to pivot independent of the rest of the airframe. I am unaware of any horizontal or vertical tail that operates in a free-wing type mode.
In the light aircraft world, plenty of designs use non fixed horizontal tails, called, "stabilators" or "all flying tails" when they first came out. With unpowered controls, you need a means of driving the surface to a particular angle for trim when not holding the stick, and a simple trim tab like you'd use on a regular elevator isn't enough. For that they use an anti-servo tab, whose range of motion relative to the stabilator can be varied by the trim control.
The anti-servo tab will drive the surface to a specific incidence and hold it there, to achieve a desired main wing trimmed angle of attack. Because of the way the tab is linked to the trim actuator, any displacement from that angle causes the tab to move in opposition to the main surface movement, driving it back to its trimmed angle. For example, if the tailing edge of the stabilator moves up from its trimmed angle, the anti-servo tab moves up relative to the trailing edge, providing an opposing (anti) aerodynamic servo restoring force that returns it to its trimmed state. Hence the name.
The anti-servo tab allows you to use the same one-piece all moving surface for stabilization, maneuvering, and trim with unpowered controls. Cessna Cardinals, and the Piper Cherokee line have such horizontal non-fixed tails, among others. Since the stabilizing surface is doing double duty as the maneuvering and trim surface, somewhat less surface area is required than a fixed stabilizer/elevator.
All moving vertical surfaces are less common in General Aviation aircraft, but what springs to mind are the Zenith and Robin designs, that use a one piece all moving rudder with no fixed stabilizer, or a very small one.
The Zeniths/Robins don't use anti-servo tabs to drive the rudder to a trimmed angle; for passive yaw stability, these airplanes rely on sufficient side area of the rear fuselage to keep the aerodynamic center of the fuselage far enough aft of the center of mass to maintain a passive weathervaning tendency.
I think fully-flying control surfaces are too common to merit a mention. Especially for supersonic airplanes they are de rigeur.
But there is one design which had a variable-sweep control surface: The Beech Starship. Here, the sweep of the canard surface could be adjusted. This helped to increase canard effectiveness in case of lowered wing flaps, but I doubt that the added weight of the sweeping mechanism could really be offset by any efficiency gain in cruise. Sweeping did, however, improve forward-down visibility.
Burt Rutan first built a 85%-scale model of the design and powered it with two PWC PT-6 turboprop engines, giving it spectacular flight performance. Once the final, much larger design was done and certified, the still two PT-6 only allowed for average performance, certainly not on the level of jets with which the Starship had to compete, and became a major (if forseeable) disappointment for Beech.
Drawing of the variable sweep canard in US patent 4,641,800.
In the end, after only 11 articles had been sold, Beech scrapped the whole program and destroyed all Starships they could get their hand on out of hatred.
Burt Ratan's SpaceShipOne is a very odd duck that counts as having "a non-fixed... stabilizer," if you mean "relative to the fuselage," or "fixed", if you mean "relative to the tail boom."
The design features a unique "feathering" atmospheric reentry system where the rear half of the wing and the twin tail booms folds 70 degrees upward along a hinge running the length of the wing; this increases drag while retaining stability.
So while the individual stabilizers are fixed to the tail boom, the tail boom itself is not fixed to the fuselage, and therefore the stabilizers change their orientation relative to the fuselage.
Here it is in the "unfeathered" configuration:
Above image downloaded from Wikipedia. By Elrondhubble, CC-BY-SA-3.0
And here it is in the "feathered" configuration:
Above image downloaded from Wikipedia. By Chris857, CC-BY-SA-3.0
MacCready Gossamer Albatross had a canard surface controlled by servotabs that actually bank left and right the whole canard wing to induce directonal control or yaw. It was free about pitch axis too.
Freedom about this banking or roll axis for canard surface is not a common thing so this may be relevant to this question.
Here are hand drawings including description and materials used.
This video shows the canard behaviour.