Some rockets actually are designed with break-away fins. Water rockets (the hobby type, not the toys) frequently use foam board fins that are designed to snap off when the rocket contacts the ground. That is especially true of lower-end models that have somewhat crude recovery systems. (Water rocketry also has produced some sophisticated designs with electronics, phased recovery system deployment, aerial photography, etc.)
In our branch of rocketry, break-away fins have been used on occasion, too. I can't recall the specifics right now, but I do remember viewing a website a couple of years ago that documented a large Level 3-type rocket project. One of the things that the author (and his partners) did was to design the fins so that they fit into brackets and were attached via a pivot at the upper end of the fin root. The rest of the root length was held in the bracket by bolts that were designed to shear off if the fin struck the ground with a certain amount of force. The fin could then pivot up on the hinge at the top so that it would not get jammed into the ground and act as a lever to produce damage to the rest of the rocket. The bolt-shearing also absorbed a bit of the impact when the rocket contacted the ground. I don't recall too much else from the site, including whether the fin attachment method worked as intended when the rocket was flown.
Mark K.