  We comparatively use some classical spectral collocation methods as well as
highly performing Chebfun algorithms in order to compute the eigenpairs of
second order singular Sturm-Liouville problems with separated self-adjoint
boundary conditions. For both the limit-circle non oscillatory and oscillatory
cases we pay a particular attention. Some "hard" benchmark problems, for which
usual numerical methods (f. d., f. e. m., etc.) fail, are analysed. For the
very challenging Bessel eigenproblem we will try to find out the source and the
meaning of the singularity in the origin. For a double singular eigenproblem
due to Dunford and Schwartz we we try to find out the precise meaning of the
notion of continuous spectrum. For some singular problems only a tandem
approach of the two classes of methods produces credible results.
