The phenomenon of gyroscopic precession is studied within the framework of the Frenet-Serret formalism adapted to quasi-Killing trajectories. Its relation to the congruence vorticity is highlighted with particular reference to the irrotational congruence admitted by the stationary, axisymmetric spacetime. General precession formulas are obtained for circular orbits with arbitrary constant angular speeds. By successive reduction, different types of precessions are derived for the Kerr-Schwarzschild-Minkowski spacetime family. The phenomenon is studied in the case of other interesting spacetimes, such as the de Sitter and Gödel universes as well as the general stationary, cylindrical, vacuum spacetimes.