In this article, we explore the phenomena of high-speed compressible boundary layer transition, laying the foundation for the development of advanced transition prediction models applicable to airflow over high-speed airfoils. Here, 2D spatial linear stability analysis (LSA) of the isothermal, compressible boundary layer over a flat plate is studied. The LSA technique is analysed by non-dimensionalisation, linearisation and Fourier-Laplace transformation of governing equations of flow. Compound matrix method (CMM) helps to remove the equation stiffness by creating all possible combinations from the original system. CMM equations are solved using the Runge–Kutta algorithm of the fourth order. Our study reveals the existence of numerous unstable modes associated with eigenvalues derived from the characteristic equations. Stability curves are obtained which highlights the complex behaviour of flow stability in the boundary layer. Moreover, the stability curves for cold and hot isothermal flat plate corresponding to both subsonic and supersonic flows are analysed. For Mach number \(M=0.6\) , the ratio of wall temperature to free stream temperature \(g_{0w} = 0.8\) and 1.4 are studied. Whereas for \(M=2.0, g_{0w}= 0.8, 1.6\) and 2.0 are studied. For the supersonic cold plate, wall temperature to free stream temperature corresponding to value 0.8, no unstable region is found till Re = 2000 and frequency( \(\omega _r\) ) ranging from 0.01 to 0.2. Notably, the critical Reynolds number exhibits an increasing trend when we transition towards colder surfaces, resulting in a reduction of the unstable area. These findings offer valuable insights into the intricate interplay of factors influencing boundary layer stability, especially concerning temperature differentials, thus contributing significantly to our understanding of transition in compressible flows.