  We study the motion of a spinning test particle in Schwarzschild spacetime,
analyzing the Poincar\'e map and the Lyapunov exponent. We find chaotic
behavior for a particle with spin higher than some critical value (e.g. $S_{cr}
\sim 0.64 \mu M$ for the total angular momentum $J=4 \mu M$), where $\mu$ and
$M$ are the masses of a particle and of a black hole, respectively. The inverse
of the Lyapunov exponent in the most chaotic case is about three orbital
periods, which suggests that chaos of a spinning particle may become important
in some relativistic astrophysical phenomena. The ``effective potential"
analysis enables us to classify the particle orbits into four types as follows.
When the total angular momentum $J$ is large, some orbits are bounded and the
``effective potential"s are classified into two types: (B1) one saddle point
(unstable circular orbit) and one minimal point (stable circular orbit) on the
equatorial plane exist for small spin; and (B2) two saddle points bifurcate
from the equatorial plane and one minimal point remains on the equatorial plane
for large spin. When $J$ is small, no bound orbits exist and the potentials are
classified into another two types: (U1) no extremal point is found for small
spin; and (U2) one saddle point appears on the equatorial plane, which is
unstable in the direction perpendicular to the equatorial plane, for large
spin. The types (B1) and (U1) are the same as those for a spinless particle,
but the potentials (B2) and (U2) are new types caused by spin-orbit coupling.
The chaotic behavior is found only in the type (B2) potential. The
``heteroclinic orbit'', which could cause chaos, is also observed in type (B2).
