We perform the linear analysis of causality and stability for a minimal extended canonical spin hydrodynamics up to second order of the gradient expansion. The first order canonical spin hydrodynamics is proved to be acausal and unstable. To remove the unstable and acausal modes, we then formulate the minimal causal spin hydrodynamics up to second order of the gradient expansion. We derive that causality and stability conditions for this minimal causal spin hydrodynamics. Interestingly, the satisfaction of the stability condition 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 linear stability conditions are necessary but may be insufficient.