  Despite recent progress, the complete understanding of the perturbations of
charged, rotating black holes as described by the Kerr-Newman metric remains an
open and fundamental problem in relativity. In this study, we explore the
existence of families of quasinormal modes of Kerr-Newman black holes whose
decay rates limit to zero at extremality, called zero-damped modes in past
studies. We review the nearly extremal and WKB approximation methods for
spin-weighted scalar fields (governed by the Dudley-Finley equation) and give
an accounting of the regimes where scalar zero-damped and damped modes exist.
Using Leaver's continued fraction method, we verify that these approximations
give accurate predictions for the frequencies in their regimes of validity. In
the nonrotating limit, we argue that gravito-electromagnetic perturbations of
nearly extremal Reissner-Nordstr\"{o}m black holes have zero-damped modes in
addition to the well-known spectrum of damped modes. We provide an analytic
formula for the frequencies of these modes, verify their existence using a
numerical search, and demonstrate the accuracy of our formula. These results,
along with recent numerical studies, point to the existence of a simple
universal equation for the frequencies of zero-damped gravito-electromagnetic
modes of Kerr-Newman black holes, whose precise form remains an open question.
