The laminar flow past a fixed sphere rotating in the transverse direction with respect to the incoming uniform flow is investigated through both direct numerical simulations (DNS) and triglobal stability analysis. The stability features of the wake are analyzed as a function of the Reynolds number Re and the sphere dimensionless rotation rate Omega. The core of the instability is precisely located by using a sensitivity analysis. Results show the existence of two different unstable regions. Moreover, an asymptotic analysis is performed to study the imperfect pitchfork bifurcation originating at Re ~ 212 and the occurrence of a steady oblique path for the problem of a freely falling sphere.