The problem of Rayleigh-Bénard convection with temperature dependent viscosity is investigated for all combinations of rigid and dynamically free boundaries. The principle of exchange of stabilities is shown to be valid for the problem which yields that the onset of convection is through stationary mode. Further, Rayleigh numbers for each possible combination of boundary conditions derived using Galerkin's technique. The effect of variable viscosity on the onset stationary convection for each case of boundary combination is computed numerically and depicted graphically. The present analysis yields that the positive values of temperature-dependent viscosity parameter has stabilizing effect on the onset of stationary convection while negative values have destabilizing effect. Further, various conclusions and results for the problem of Rayleigh-Bénard convection with constant viscosity for different cases of boundary conditions have been worked out and are discussed in detail.