  We analyze the collider signatures of the real singlet extension of the
Standard Model in regions consistent with a strong first-order electroweak
phase transition and a singlet-like scalar heavier than the Standard Model-like
Higgs. A definitive correlation exists between the strength of the phase
transition and the trilinear coupling of the Higgs to two singlet-like scalars,
and hence between the phase transition and non-resonant scalar pair production
involving the singlet at colliders. We study the prospects for observing these
processes at the LHC and a future 100 TeV $pp$ collider, focusing particularly
on double singlet production. We also discuss correlations between the strength
of the electroweak phase transition and other observables at hadron and future
lepton colliders. Searches for non-resonant singlet-like scalar pair production
at 100 TeV would provide a sensitive probe of the electroweak phase transition
in this model, complementing resonant di-Higgs searches and precision
measurements. Our study illustrates a strategy for systematically exploring the
phenomenologically viable parameter space of this model, which we hope will be
useful for future work.
