We investigate the dynamical stability and instability for Rayleigh–Bénard problem of three-dimensional incompressible non-Newtonian fluids with Bingham-type. In this paper, we devote to establishing a stability criterion \(\frac{1}{R}>\sup _{(\varvec{u}, \theta )\in \mathcal {A}}\frac{2\int u_3\theta \textrm{d}x}{\Vert (\nabla \varvec{u}, \nabla \theta )\Vert ^2_0}\) and proving that the Rayleigh–Bénard problem is exponential stable by energy method under this criterion after overcoming the difficulty caused by the nonlinear term in the mathematical model of Bingham fluid. Moreover, we provide an instability criterion \(\frac{1}{R}<\sup _{(\varvec{u}, \theta )\in \mathcal {A}} \frac{2\int u_3\theta \textrm{d}x}{\Vert (\nabla \varvec{u}, \nabla \theta )\Vert ^2_0}\) , then we will prove that the Rayleigh–Bénard problem is unstable under this instability criterion by using the energy method and the modified variational method. Our result shows that non-Newtonian part of this fluid has the stabilizing effect for thermal instability.
