  So far analysis of the quasinormal spectrum of a massive charged scalar field
in the black hole background has been limited by the regime of small \mu M and
qQ, where \mu, q (M, Q) are mass and charge of the field (black hole). Here we
shall present a comprehensive picture of quasinormal modes, late-time tails and
stability of a massive charged scalar field around Kerr-Newman black holes for
any physically meaningful values of the parameters. We shall show that despite
presence of the two mechanisms of superradiance (owing to black hole's rotation
and charge) and the massive term creating growing bound states, there is no
indication of instability under quasinormal modes' boundary conditions. We have
shown that for some moderate values of qQ dominant quasinormal modes may have
arbitrarily small real oscillation frequencies Re(\omega). An analytic formula
for the quasinormal modes has been derived in the regime of large qQ. The
larger the field's charge, the sooner asymptotic tails dominate in a signal,
making it difficult to extract quasinormal frequencies from a time-domain
profile. Analytic expressions for intermediate and asymptotically late-time
tails have been found for the Reissner-Nordstr\"om black hole. For the near
extremal Kerr-Newman black holes we have obtained a more general picture of the
mode branching found recently for massless fields [arXiv:1212.3271] in the Kerr
background.
