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 $h rightarrow gammagamma$ and $h rightarrow Zgamma$, 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 $h rightarrow Zgamma$ 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.
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