Steady thermal convection of an infinite Prandtl number, Boussinesq fluid with temperature-dependent viscosity is systematically examined in a three-dimensional, basally heated spherical shell with isothermal and stress-free boundaries. Convective flows exhibiting cubic (l=2, m={0,4}) and tetrahedral (l=3, m=2) symmetry are generated with a finite-volume numerical model for various combinations of Rayleigh number Ra (defined with viscosity based on the average of the boundary temperatures) and viscosity contrast rμ (ratio of maximum to minimum viscosities). The range of Ra for which these symmetric flows in spherical geometry can be maintained in steady state is sharply reduced by even mild viscosity variations (rμ≤30), in contrast with analogous calculations in Cartesian geometry in which relatively simple, three-dimensional convective planforms remain steady for rμ≈104. The mild viscosity contrasts employed place some solutions marginally in the sluggish-lid transition regime in Ra-rμ parameter space. Global heat transfer, given by the Nusselt number Nu, is found to obey a single power law relation with Ra when Ra is scaled by its critical value. A power law of the form Nu~(Ra/Racrit)1/4 (Racrit is the minimum critical value of Ra for the onset of convection) is obtained, in agreement with previous results for isoviscous spherical shell convection with cubic and tetrahedral symmetry. The calculations of this paper demonstrate that temperature dependent viscosity exerts a strong control on the nature of three-dimensional convection in spherical geometry, an effect that is likely to be even more important at Rayleigh numbers and viscosity contrasts more representative of the mantles of terrestrial planets. The robustness of the Nu-Ra relation, when scaled by Racrit, is important for studies of planetary thermal history that rely on parameterizations of convective heat transport and account for temperature dependence of mantle viscosity. ¿ American Geophysical Union 1996