  We perform a thorough scan of the parameter space of a general singlet scalar
extension of the Standard Model to identify the regions which can lead to a
strong first-order phase transition, as required by the electroweak
baryogenesis mechanism. We find that taking into account bubble nucleation is a
fundamental constraint on the parameter space and present a conservative and
fast estimate for it to enable efficient parameter space scanning. The allowed
regions turn out to be already significantly probed by constraints on the
scalar mixing from Higgs signal strength measurements. We also consider the
addition of new neutrino singlet fields with Yukawa couplings to both scalars
and forming heavy (pseudo)-Dirac pairs, as in the linear or inverse see-saw
mechanisms for neutrino mass generation. Interestingly, we identify an
interplay between the strength of the phase transition and the stability of the
electroweak vacuum which prevents these Yukawa couplings to become arbitrarily
large. Thus, their inclusion does not alter the early universe phenomenology or
allowed parameter space in a significant way. Conversely, we find allowed
regions of the parameter space where the presence of the neutrino singlets
would remarkably modify the collider phenomenology, yielding interesting new
signatures in Higgs and singlet scalar decays.
