  The Mathisson-Papapetrou-Dixon (MPD) equations describe the motion of
spinning test particles. It is well-known that these equations, which couple
the Riemann curvature tensor with the antisymmetric spin tensor S, together
with the normalization condition for the four-velocity, is a system of eleven
equations relating fourteen unknowns. To ``close'' the system, it is necessary
to introduce a constraint of the form V_\mu S^{\mu \nu} = 0, usually known as
the spin supplementary condition (SSC), where V_\mu is a future-oriented
reference vector satisfying the normalization condition V_\alpha V^\alpha = -1.
There are several SSCs in the literature. In particular, the Tulzcyjew-Dixon,
Mathisson-Pirani, and Ohashi-Kyrian-Semer\'ak are the most used by the
community. From the physical point of view, choosing a different SSC (a
different reference vector $V^\mu$) is equivalent to fixing the centroid of the
test particle. In this manuscript, we compare different SSCs for spinning test
particles moving around a Morris-Thorne traversable wormhole. To do so, we
first obtain the orbital frequency and expand it up to third-order in the
particle's spin; as expected, the zero-order coincides with the Keplerian
frequency, the same in all SSCs; nevertheless, we found that differences appear
in the second order of the expansion, similar to the Schwarzschild and Kerr
black holes. We also compare the behavior of the innermost stable circular
orbit (ISCO). Since each SSC is associated with a different centroid of the
test particle, we analyze (separately) the radial and spin corrections for each
SSC. We found that the radial corrections improve the convergence, especially
between Tulzcyjew-Dixon and Mathisson-Pirani SSCs. In the case of
Ohashi-Kyrian-Semer\'ak, we found that the spin corrections remove the
divergence for the ISCO and extend its existence for higher values of the
particle's spin.
