(Abridged) We analyse the stability and evolution of power-law accretion disc models. These have midplane densities that follow radial power-laws, and have either temperature or entropy distributions that are power-law functions of cylindrical radius. We employ two different hydrodynamic codes to perform 2D-axisymmetric and 3D simulations that examine the long-term evolution of the disc models as a function of the power-law indices of the temperature or entropy, the thermal relaxation time of the fluid, and the viscosity. We present a stability analysis of the problem that we use to interpret the simulation results. We find that disc models whose temperature or entropy profiles cause the equilibrium angular velocity to vary with height are unstable to the growth of modes with wavenumber ratios |k_R/k_Z| >> 1 when the thermodynamic response of the fluid is isothermal, or the thermal evolution time is comparable to or shorter than the local dynamical time scale. These discs are subject to the Goldreich-Schubert-Fricke (GSF) or `vertical shear linear instability. Development of the instability involves excitation of vertical breathing and corrugation modes in the disc, with the corrugation modes in particular being a feature of the nonlinear saturated state. Instability operates when the dimensionless disc kinematic viscosity nu < 10^{-6} (Reynolds numbers Re>H c_s/nu > 2500). In 3D the instability generates a quasi-turbulent flow, and the Reynolds stress produces a fluctuating effective viscosity coefficient whose mean value reaches alpha ~ 6 x 10^{-4} by the end of the simulation. The vertical shear instability in disc models which include realistic thermal physics has yet to be examined. Should it occur, however, our results suggest that it will have significant consequences for their internal dynamics, transport properties, and observational appearance.
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