  This work investigates a simple, representative extension of the Standard
Model with a real scalar singlet and spontaneous $Z_2$ breaking, which allows
for a strongly first-order phase transition, as required by electroweak
baryogenesis. We perform analytical and numerical calculations that
systematically include one-loop thermal effects, Coleman-Weinberg corrections,
and daisy resummation, as well as evaluation of bubble nucleation. We study the
rich thermal history and identify the conditions for a strongly first-order
electroweak phase transition with nearly degenerate extrema at zero
temperature. This requires a light scalar with mass below 50 GeV. Exotic Higgs
decays, as well as Higgs coupling precision measurements at the LHC and future
collider facilities, will test this model. Additional information may be
obtained from future collider constraints on the Higgs self-coupling.
Gravitational-wave signals are typically too low to be probed by future
gravitational wave experiments.
