  We study the non-Euclidean (incompatible) elastic energy functionals in the
description of prestressed thin films, at their singular limits
($\Gamma$-limits) as $h\to 0$ in the film's thickness $h$. Firstly, we extend
the prior results [Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner,
Lewicka-Raoult-Ricciotti] to arbitrary incompatibility metrics that depend on
both the midplate and the transversal variables (the "non-oscillatory" case).
Secondly, we analyze a more general class of incompatibilities, where the
transversal dependence of the lower order terms is not necessarily linear (the
"oscillatory" case), extending the results of [Agostiniani-Lucic-Lucantonio,
Schmidt] to arbitrary metrics and higher order scalings. We exhibit connections
between the two cases via projections of appropriate curvature forms on the
polynomial tensor spaces. We also show the effective energy quantisation in
terms of scalings as a power of $h$ and discuss the scaling regimes $h^2$
(Kirchhoff), $h^4$ (von K\'arm\'an) in the general case, as well as all
possible (even powers) regimes for conformal metrics, thus paving the way to
the subsequent complete analysis of the non-oscillatory setting in [Lewicka].
Thirdly, we prove the coercivity inequalities for the singular limits at $h^2$-
and $h^4$- scaling orders, while disproving the full coercivity of the
classical von K\'arm\'an energy functional at scaling $h^4$.
