External forces provide a means of actively manipulating suspended particles in microscale flows, but their coupling with inertial migration and non-Newtonian rheology remains poorly understood. In particular, how streamwise and cross-stream forces modify the force balance governing parti- cle equilibria in shear-thinning fluids, how the resulting equilibria depend on the flow inertia, and whether additional migration states can emerge, have received limited attention. Here, we inves- tigate the inertial migration of a neutrally buoyant rigid sphere in plane Couette flow of a shear- thinning Carreau fluid subjected independently to streamwise and cross-stream external forces. Three-dimensional simulations are performed using a smoothed-profile lattice Boltzmann method, with the local viscosity updated from the Carreau constitutive model. In the absence of forcing, the particle undergoes the previously established supercritical pitchfork bifurcation from a stable centreline equilibrium to symmetric off-centre equilibria, while increasing the Carreau number re- duces the critical channel Reynolds number. External forcing breaks the symmetry and modifies the equilibrium through distinct mechanisms. Streamwise forcing generates particle–fluid slip and an associated Saffman-type inertial lift, producing a force-dependent displacement of the equilibrium towards the wall. Cross-stream forcing acts directly in the migration direction and displaces the equilibrium away from the centreline, with the direction of displacement determined by the sign of the applied force; in this case, the equilibrium position depends non-monotonically on the channel Reynolds number. Force-balance analysis shows that the equilibrium positions and their stability are governed by the competition between the imposed force and the hydrodynamic lift, with shear thinning modifying this balance through spatial variations in the effective viscosity. In particular, the viscosity field exhibits a pronounced reduction in the region of high shear surrounding the par- ticle, which modifies the local hydrodynamic forces and enhances the response to external forcing. At sufficiently high Reynolds numbers and weak forcing, an additional equilibrium emerges in the opposite half of the channel, leading to coexistence of two stable, non-mirror-symmetric equilibria. The associated saddle-node structure constitutes an imperfect bifurcation, with its onset occurring at lower Reynolds numbers as shear thinning increases. Finally, a rotation-arrested control cal- culation shows that the rotation-induced contribution remains small compared with the slip- and wall-induced contributions to the transverse lift over the parameter range considered. These results provide a framework for understanding and controlling force-driven inertial migration in confined shear-thinning flows.