Abstract The Rayleigh–Bénard thermoconvection of Newtonian fluids has been extensively studied. The transition from the conductive, static state to thermoconvection flows corresponds in this case to a supercritical bifurcation. In shear-thinning fluids, on the contrary, recent weakly nonlinear studies have shown that the transition may become subcritical. Using a custom numerical code developed with Freefem++ to compute bidimensional, fully nonlinear roll solutions in Carreau fluids, for a large range of rheological parameters, and more particularly for strongly shear-thinning fluids, approaching power-law fluids, we confirm this result. A simple expression of the value of the Rayleigh number at which subcritical convection rolls appear is proposed. This law suggests to reconsider the choice of the reference viscosity for shear-thinning fluids. Indeed, when the shear-thinning effects increase, the critical Rayleigh number increases or decreases depending on the choice of the reference viscosity. The ‘neutral’ or ‘effective’ viscosity, which gives a constant value of the Rayleigh number at the onset of subcritical convection rolls, is close to the bulk average viscosity. In addition, a correlation is proposed to estimate the Nusselt number of subcritical rolls.