Scalar dissipation rate plays an important role in combustion, especially for fast chemistry in non-premixed flames. Even for finite-rate chemistry, it is a parameter of major importance in models based on flamelets, probability density function methods, and conditional moment closures. Compressibility can influence the behavior of scalar dissipation but existing models do not consider them in the form of large dilatational effects, which are present in combustion in high-speed propulsion systems such as scramjets in hypersonic flight. This paper considers these effects on the scalar dissipation rate, in particular its mean value, spectral density, probability density function, as well the alignment of the scalar gradient with the strain field of the flow. The major result of this study is that the majority of the compressibility effects on inertial-type scales can be taken into account if the classical scaling laws of incompressible turbulence are rescaled by taking only the solenoidal contributions. However, additional effects of compressibility need to be considered at smaller scales.