  The $U(1)$ gauge-invariant scalar-vector-tensor theories, which catches five
degrees of freedom, are valuable for its implications to inflation problems,
generation of primordial magnetic fields, new black hole (BH) and neutron star
solutions, etc. In this paper, we derive conditions for the absence of ghosts
and Laplacian instabilities of nontrivial BH solutions dressed with scalar hair
against both odd- and even-parity perturbations on top of the static and
spherically symmetric background in the most general $U(1)$ gauge-invariant
scalar-vector-tensor theories with second-order equations of motion. In
addition to some general discussions, several typical concrete models are
investigated. Specially, we show that the stability against even-parity
perturbations is ensured outside the event horizon under certain constraints to
these models. This is a crucial step to check the self-consistency of the
theories and to shed light on the physically accessible models of such theories
for future studies.
