  We establish asymptotic estimates of Mathieu-type series defined by sequences
with power-logarithmic or factorial behavior. By taking the Mellin transform,
the problem is mapped to the singular behavior of certain Dirichlet series,
which is then translated into asymptotics for the original series. In the case
of power-logarithmic sequences, we obtain precise first order asymptotics. For
factorial sequences, a natural boundary of the Mellin transform makes the
problem more challenging, but a direct elementary estimate gives reasonably
precise asymptotics.
