  We perform the linear analysis of causality and stability for a minimal
extended spin hydrodynamics up to second order of the gradient expansion. The
first order spin hydrodynamics, with a rank-3 spin tensor being antisymmetric
for only the last two indices, are proved to be acausal and unstable. We then
consider the minimal causal spin hydrodynamics up to second order of the
gradient expansion. We derive the necessary causality and stability conditions
for this minimal causal spin hydrodynamics. Interestingly, the satisfaction of
the stability conditions relies on the equations of state for the spin density
and chemical potentials. Moreover, different with the conventional relativistic
dissipative hydrodynamics, the stability of the theory seems to be broken at
the finite wave-vector when the stability conditions are fulfilled at small and
large wave-vector limits. It implies that the behavior in small and large
wave-vector limits may be insufficient to determine the stability conditions
for spin hydrodynamics in linear mode analysis.
