  The local gravitational instability of rotating discs is believed to be an
important mechanism in different astrophysical processes, including the
formation of gas and stellar clumps in galaxies. We aim to study in three
dimensions the local gravitational instability of two-component thick discs. We
take as starting point a recently proposed analytic three-dimensional (3D)
instability criterion for discs with non-negligible thickness which takes the
form $Q_{\rm 3D}<1$, where $Q_{\rm 3D}$ is a 3D version of the classical 2D
Toomre $Q$ parameter for razor-thin discs. Here we extend the 3D stability
analysis to two-component discs, considering first the influence on $Q_{\rm
3D}$ of a second unresponsive component, and then the case in which both
components are responsive. We present the application to two-component discs
with isothermal vertical distributions, which can represent, for instance,
galactic discs with both stellar and gaseous components. Finally, we relax the
assumption of vertical isothermal distribution, by studying one-component
self-gravitating discs with polytropic vertical distributions for a range of
values of the polytropic index corresponding to convectively stable
configurations. We find that $Q_{\rm 3D}<1$, where $Q_{\rm 3D}$ can be computed
from observationally inferred quantities, is a robust indicator of local
gravitational instability, depending only weakly on the presence of a second
component and on the vertical gradient of temperature or velocity dispersion.
We derive a sufficient condition for local gravitational instability in the
midplane of two-component discs, which can be employed when both components
have $Q_{\rm 3D}>1$.
