  We consider the motion of spinning test particles with nonzero rest mass in
the "pole-dipole" approximation, as described by the Mathisson-Papapetrou-Dixon
(MPD) equations, and examine its properties in dependence on the spin
supplementary condition added to close the system. The MPD equation of motion
is decomposed in the orthonormal tetrad whose time vector is given by the
four-velocity $V^\mu$ chosen to fix the spin condition (the "reference
observer") and the first spatial vector by the corresponding spin; such
projections do not contain the Weyl scalars $\Psi_0$ and $\Psi_4$ obtained in
the associated Newman-Penrose (NP) null tetrad. One natural choice of the
remaining two spatial basis vectors is shown to follow "intrinsically"; it is
realizable if the particle's four-velocity and four-momentum are not parallel.
To see how the problem depends on the curvature type, one first identifies the
first vector of the NP tetrad $k^\mu$ with the highest-multiplicity principal
null direction of the Weyl tensor, and then sets $V^\mu$ so that $k^\mu$ belong
to the spin-bivector eigenplane. In spacetimes of any algebraic type but III,
it is possible to rotate the tetrads so as to become "transverse", namely so
that $\Psi_1$ and $\Psi_3$ vanish. If the spin-bivector eigenplane could be
made coincide with the real-vector plane of any of such transverse frames, the
motion would consequently be fully determined by $\Psi_2$ and the cosmological
constant; however, this can be managed in exceptional cases only. Besides
focusing on specific Petrov types, we derive several sets of useful relations
valid generally and check whether/how the exercise simplifies for some specific
types of motion. The particular option of having four-velocity parallel to
four-momentum is advocated and a natural resolution of nonuniqueness of the
corresponding reference observer $V^\mu$ is suggested.
