We study the motion of spinning test particles in Kerr spacetime using the Mathisson–Papapetrou equations; we impose different supplementary conditions from the well-known conditions of Corinaldesi–Papapetrou, Pirani and Tulczyjew, and analyse their physical implications in order to decide which is the most natural to use. We find that if the particle's centre-of-mass world line, namely the one chosen for multipole reduction, is a spatially circular orbit (sustained by the tidal forces due to the spin), then the generalized momentum P of the test particle is also tangent to a spatially circular orbit intersecting the centre-of-mass line at a point. There exists one such orbit for each point of the centre-of-mass line where they intersect; although fictitious, these orbits are essential to define the properties of the spinning particle along its physical motion. In the small spin limit, the particle orbit is almost a geodesic and the difference in its angular velocity with respect to the geodesic value can be of arbitrary sign, corresponding to the possible spin-up and -down alignments along the z-axis. We also find that the choice of the supplementary conditions leads to clock effects of substantially different magnitude. In fact, for co- and counter-rotating particles having the same spin magnitude and orientation, the gravitomagnetic clock effect induced by the background metric can be magnified or inhibited and even suppressed by the contribution of the spin of individual particles. Quite surprisingly, this contribution can itself be made vanishingly small, leading to a clock effect indistiguishable from that of non-spinning particles. The results of our analysis can be observationally tested.