This dissertation focuses on the study of spectral stability in traveling waves, with a special interest in planar detonations in the multidimensional reactive Navier-Stokes equations. The chief tool is the Evans function, combined with STABLAB, a numerical library devoted to calculating the Evans function. Properly constructed, the Evans function is an analytic function in the right half-plane whose zeros correspond in multiplicity and location to the spectrum of the traveling wave. Thus the Evans function can be used to verify stability, or to locate precisely any unstable eigenvalues. We introduce a new method that uses numerical continuation to follow unstable eigenvalues as system parameters vary. We also use the Evans function to track instabilities of viscous detonations in the multidimensional reactive Navier-Stokes equations, building on recent results for detonations in one dimension. Finally, we introduce a Python implementation of STABLAB, which we hope will improve the accessibility of STABLAB and aid the future study of large, multidimensional systems by providing easy-to-use parallel processing tools.