We explore thermal convection of a fluid with a temperature-dependent viscosity in a basally heated 3-D spherical shell using linear stability analyses and numerical experiments, while considering the application of our results to terrestrial planets. The inner to outer radius ratio of the shell f assumed in the linear stability analyses is in the range of 0.11-0.88. The critical Rayleigh number R-c for the onset of thermal convection decreases by two orders of magnitude as f increases from 0.11 to 0.88, when the viscosity depends sensitively on the temperature, as is the case for real mantle materials. Numerical simulations carried out in the range of f = 0.11-0.55 show that a thermal boundary layer (TBL) develops both along the surface and bottom boundaries to induce cold and hot plumes, respectively, when f is 0.33 or larger. However, for smaller f values, a TBL develops only on the bottom boundary. Convection occurs in the stagnant-lid regime where the root mean square velocity on the surface boundary is less than 1 per cent of its maximum at depth, when the ratio of the viscosity at the surface boundary to that at the bottom boundary exceeds a threshold that depends on f. The threshold decreases from 10(6.5) at f = 0.11 to 10(4) at f = 0.55. If the viscosity at the base of the convecting mantle is 10(20)-10(21) Pa s, the Rayleigh number exceeds R-c for Mars, Venus and the Earth, but does not for the Moon and Mercury; convection is unlikely to occur in the latter planets unless the mantle viscosity is much lower than 10(20) Pa s and/or the mantle contains a strong internal heat source.