  We study the stability of first-order dissipative spin-hydrodynamic
frameworks. We considered two different first-order dissipative
spin-hydrodynamic frameworks. The first one considers the spin chemical
potential ($\omega^{\alpha\beta}$) to be first order ($\mathcal{O}(\partial)$)
in the hydrodynamic gradient expansion. The hydrodynamic gradient ordering of
the spin chemical potential is a debatable issue within the frameworks of spin
hydrodynamics. Therefore as a second choice, we also consider the spin
hydrodynamic equations with $\omega^{\alpha\beta}\sim\mathcal{O}(1)$. We find
that for both frameworks, at the level of linear perturbations some spin modes
can be unstable. To remove these generic instabilities we consider the Frenkel
condition. We argue that Frenkel condition helps get rid of the unstable
solutions in both cases, but with a physical drawback for the case where
$\omega^{\mu\nu}\sim\mathcal{O}(\partial)$.
