  In this article we study quantitative rigidity properties for the compatible
and incompatible two-state problems for suitable classes of $\mathcal{A}$-free
operators and for a singularly perturbed $T_3$-structure for the divergence
operator. In particular, in the compatible setting of the two-state problem we
prove that all homogeneous, first order, linear operators with affine boundary
data which enforce oscillations yield the typical
$\epsilon^{\frac{2}{3}}$-lower scaling bounds. As observed in \cite{CC15} for
higher order operators this may no longer be the case. Revisiting the example
from \cite{CC15}, we show that this is reflected in the structure of the
associated symbols and that this can be exploited for a new Fourier based proof
of the lower scaling bound. Moreover, building on \cite{RT22, GN04, PP04}, we
discuss the scaling behaviour of a $T_3$ structure for the divergence operator.
We prove that as in \cite{RT22} this yields a non-algebraic scaling law.
