  We study transcendental singularities of a Schr\"oder map arising from a
rational function $f$, using results from complex dynamics and Nevanlinna
theory. These maps are transcendental meromorphic functions of finite order in
the complex plane. We show that their transcendental singularities lie over the
set where $f$ is not semihyperbolic (unhyperbolic). In addition, if they are
direct, then they lie over only attracting periodic points of $f$, and
moreover, if $f$ is a polynomial, then both direct and indirect singularities
lie over attracting, parabolic and Cremer periodic points of $f$. We also
obtain concrete examples of both kinds of transcendental singularities of
Schr\"oder maps as well as a new proof of the Pommerenke-Levin-Yoccoz
inequality and a new formulation of the Fatou conjecture.
