The flow over a rotating cone in still fluid is susceptible to crossflow and centrifugal instability modes of spiral nature, depending on the cone half-angle. For parameters ranging from propeller nose cones to rotating disks, the instability triggers co-rotating vortices, whereas for slender spinning missiles, counter-rotating vortices are observed. Upon introduction of an oncoming flow, the problem essentially becomes a battle between the streamwise and azimuthal shear flow, due to the rotating surface. The boundary layer instability is again visualized by the formation of spiral vortices, which wrap around the cone surface in a helical nature. For both crossflow and centrifugal instabilities, we derive the mean flow boundary layer equations and investigate the high Reynolds number asymptotic linear stability of the flow to inviscid crossflow modes (type I), type II modes, which arise from a viscous-Coriolis force balance, and neutral modes for a slender cone. The influence of the cone half-angle (ψ) and axial flow strength (s or Ts) on the number and orientation of the spiral vortices is examined, with comparisons made with previous experimental and numerical results.