We investigate the influence of rotational and vibrational energy relaxation on the stability of laminar boundary layers in supersonic flows by numerically solving the linearized equations of motion for a flow in thermal non-equilibrium. We model air as a mixture of nitrogen, oxygen and carbon dioxide, and derive accurate models for the relaxation rates from published experimental data in the field of physical chemistry. The influence of rotational relaxation is to dampen high-frequency instabilities, consistent with the well known damping effect of rotational relaxation on acoustical waves. The influence of rotational relaxation can be modelled with acceptable accuracy through the use of the bulk-viscosity approximation when the bulk viscosity is computed with a formula described herein. Vibrational relaxation affects the growth of disturbances by changing the characteristics of the laminar mean flow. The influence is strongest when the flow field contains a region at, or near, stagnation conditions, followed by a rapid expansion, such as inside wind tunnels and around bodies with a blunt leading edge, whereby the rapid expansion causes the internal energy to freeze in a distribution out of equilibrium. For flows at Mach 4.5 and stagnation temperature of 1000 K, the total amplification exhibited by boundary-layer disturbances over a sharp flat plate in wind-tunnel flows can reach a value that is fifty times as high as the value computed under the assumption of thermal equilibrium. The difference in amplification can be twice as high in the case of a blunt flat plate at atmospheric flight conditions.