  The inert doublet model is a well-motivated extension of the Standard Model
that contains a dark matter candidate and modifies the dynamics of the
electroweak symmetry breaking. In order to detail its phenomenology, we perform
a comprehensive study of cosmic phase transitions and gravitational wave
signals implied by the framework, accounting for the latest results of collider
experiments. We require the neutral inert scalar to constitute, at least, a
subdominant part of the observed dark matter abundance. While most of the phase
transitions proceed through a single step, we identify regions of the parameter
space where the electroweak vacuum is reached after multiple phase transitions.
The resulting gravitational wave spectrum is generally dominated by single-step
transitions and, in part of the parameter space, falls within the reach of
future gravitational wave detectors such as LISA, BBO or DECIGO. We find that
direct detection experiments efficiently probe the part of parameter space
associated with multi-step phase transitions, which remain unconstrained only
in the Higgs resonance region testable with future monojet searches. The
implications of the new determination of the $W$ boson mass are also discussed.
