Shear flows have an important impact on the dynamics in an assortment of different astrophysical objects including accreditation discs and stellar interiors. Investigating shear flow instabilities in a polytropic atmosphere provides a fundamental understanding of the motion in stellar interiors where turbulent motions, mixing processes, as well as magnetic field generation takes place. Here, a linear stability analysis for a fully compressible fluid in a two-dimensional Cartesian geometry is carried out. Our study focuses on determining the critical Richardson number for different Mach numbers and the destabilising effects of high thermal diffusion. We find that there is a deviation of the predicted stability threshold for moderate Mach number flows along with a significant effect on the growth rate of the linear instability for small Peclet numbers. We show that in addition to a Kelvin-Helmholtz instability a Holmboe instability can appear and we discuss the implication of this in stellar interiors.