The linear stability of free-shear flows is governed by their dispersion characteristics. The dispersion relation can be obtained by integrating the Rayleigh equation. The integration process can be hampered by the presence of singularities within the domain of integration. A complex-domain contour integration procedure is presented that enables this integration to be performed in a modular and robust fashion. This is accomplished by deforming the original integration contour into piecewise-continuous line-segments in the complex domain to avoid all the singularities. This integration technique can then be used to find absolute and convective instabilities of the medium by a simple procedure. However when the velocity profile for a shear layer is obtained from experiments or numerical simulations, it is available only along the real-axis. Thus the complex-domain integration procedure cannot be applied unless a functional fit is obtained for the velocity profile. For convectively unstable systems, the integration can be carried out along the real-axis only for self-excited systems. However, for a certain class of free-shear flows, it is shown that an absolute instability can still be calculated by integrating the Rayleigh equation along the real-axis. This leads to the development of a fully automatic absolute-instability solver and a semi-automatic convective-instability solver.