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. © 2014, Springer-Verlag Berlin Heidelberg.