An efficient linearized compressible Navier-Stokes solver has been developed to conduct laminar-turbulent transition predictions in hypersonic flows. Numerical Jacobians are employed to avoid lengthy, error prone derivation and implementation of the stability equations. Combined with a generalized curvilinear implementation, the approach is directly applicable for stability investigations of complex geometries. The governing equations are discretized in time using time-stepping and time-spectral schemes. The time-spectral method allows to efficiently solve time periodic problems by completely bypassing the initial transients that are often not of interest. In this paper, validation results obtained with the time-stepping and time-spectral schemes are presented for an incompressible temporal shear layer, a supersonic spatial shear layer, hypersonic boundary layers on a at plate and cones (straight and ared) and for a biglobal stability analysis of a cylinder wake. Finally, preliminary results for a cone at angle of attack which is susceptible to cross-flow instability are presented.