The motion of non-neutrally buoyant prolate spheroidal particles in vertical shear flows is investigated. Using the generalized Faxen law, we calculate the hydrodynamic forces and moments acting on such inertial and inertialess particles, and their trajectories. The calculations are done for (i) freely rotating particles, and (ii) particles with orientations fixed by means of an external torque exerted by a strong orienting field. Inertial particles are found to migrate across the streamlines, and their trajectories differ considerably from those calculated for inertialess particles. Neutrally buoyant spheroids, inertial or not, which either freely rotate or have fixed orientations in shear flows, translate along the streamlines. Non-neutrally buoyant inertialess spheroids freely moving in simple shear flow translate along periodic trajectories with no net lateral drift. In contrast, inertial particles under similar flow conditions drift laterally toward locations characterized by higher local velocities in a direction opposing gravity. The motion of non-neutrally buoyant inertial particles with fixed orientations may be unstable with the drift velocity growing exponentially with time. Conditions for the occurrence of this unstable motion are formulated analytically in terms of particle and flow parameters. In general, the rate of drift depends on particle shape, via its aspect ratio, and its inertia.