It was shown by Ruell and Takens that fluid motion can be chaotic with three distinct frequencies being aquired due to instabilities of flow. Thus, the instability of a steady flow is the first step to chaos and hence to turbulence. After overviewing the scenario of transitions of flow to turbulence briefly, we show four numerical methods to analyze the linear stability of steady flows: numerical simulation of a set of linear equations for a disturbance, successive over relaxation method, Arnoldi method, and shift and inverse power method. We discuss the merit of each numerical method together with some issues arising in numerical stability analyses of flow past a circular cylinder, taking it as an example.