We study the behaviour of spinning test particles in the Schwarzschild spacetime. Using Mathisson–Papapetrou equations of motion, we confine our attention to spatially circular orbits and search for observable effects which could eventually discriminate among the standard supplementary conditions, namely the Corinaldesi–Papapetrou, Pirani and Tulczyjew. We find that if the world line chosen for the multipole reduction and whose unit tangent we denote as U is a circular orbit then the generalized momentum P of the spinning test particle is also tangent to a circular orbit even though P and U are not parallel four-vectors. These orbits are shown to exist because the spin-induced tidal forces provide the required acceleration irrespective of the supplementary conditions we select. Of course, in the limit of a small spin, the particle's orbit is close to being a circular geodesic and the (small) deviation of the angular velocities from the geodesic values can be of an arbitrary sign, corresponding to the possible spin-up and spin-down alignment to the z-axis. When two spinning particles orbit around a gravitating source in opposite directions, they make one loop with respect to a given static observer with different arrival times. This difference is termed the clock effect. We find that a nonzero gravitomagnetic clock effect appears for oppositely orbiting spin-up or spin-down particles even in the Schwarzschild spacetime. This allows us to establish a formal analogy with the case of (spin-less) geodesics on the equatorial plane of the Kerr spacetime. This result can be verified experimentally.