A test gyroscope is a point-like massive particle having an additional “structure” termed spin. A spinning body is not, strictly speaking, point-like because its average size cannot be less than the ratio between its spin and its mass, in geometrized units. This guarantees that no point of the spinning body moves with respect to any observer at a velocity larger than c.
Rotation is a common feature in the universe so knowing the dynamics of rotating bodies is almost essential in modern physics. Although in most cases the assumption of no rotation is necessary to make the equations tractable, the existence of a strictly non-rotating system should be considered a rare event, and a measurement which revealed one would be of great interest. This is the case for the massive black holes which appear to exist in the nuclei of most galaxies. Detailed measurements are aimed at detecting their intrinsic angular momentum from the behavior of the surrounding medium. A black hole, however, could also be seen directly if we were able to detect the gravitational radiation it would emit after being perturbed by an external field. A direct measurement of gravitational waves is still out of reach for earthbound detectors; nevertheless they are still extensively searched for since they are the ultimate resource to investigate the nature of space-time.