  The available data on the 125 GeV scalar $h$ is analysed to explore the room
for new physics in the electroweak symmetry breaking sector. The first part of
the study is model-independent, with $h$ couplings to standard model particles
scaled by quantities that are taken to be free parameters. At the same time,
the additional loop contributions to $h \rightarrow \gamma\gamma$ and $h
\rightarrow Z\gamma$, mediated by charged scalar contributions in the extended
scalar sector, are treated in terms of gauge-invariant effective operators.
Having justified this approach for cases where the concerned scalar masses are
a little above the $Z$-boson mass, we fit the existing data to obtain
marginalized 1$\sigma$ and 2$\sigma$ regions in the space of the coefficients
of such effective operators, where the limit on the $h \rightarrow Z\gamma$
branching ratio is used as a constraint. The correlation between, say, the
gluon fusion and vector-boson fusion channels, as reflected in a non-diagonal
covariance matrix, is taken into account. After thus obtaining
model-independent fits, the allowed values of the coefficients are translated
into permissible regions of the parameter spaces of several specific models. In
this spirit we constrain four different types of two Higgs doublet models, and
also models with one or two $Y = 2$ scalar triplets, taking into account the
correlatedness of the scale factors in $h$-interactions and the various
couplings of charged Higgs states in each extended scenario.
