In this paper we develop approximations to the characteristic roots of delay differential
equations using the spectral tau and spectral least squares approach. We study the
influence of different choices of basis functions in the spectral solution on the
numerical convergence of the characteristic roots. We found that the spectral tau
method performed better than the spectral least squares method. Legendre and Chebyshev
bases provide much better convergence properties than the mixed Fourier basis.