  Scalar field theories with $\mathbb{Z}_{2}$-symmetry are the traditional
playground of critical phenomena. In this work these models are studied using
functional renormalization group (FRG) equations at order $\partial^2$ of the
derivative expansion and, differently from previous approaches, the spike plot
technique is employed to find the relative scaling solutions in two and three
dimensions. The anomalous dimension of the first few universality classes in
$d=2$ is given and the phase structure predicted by conformal field theory is
recovered (without the imposition of conformal invariance), while in $d=3$ a
refined view of the standard Wilson-Fisher fixed point is found. Our study
enlightens the strength of shooting techniques in studying FRG equations,
suggesting them as candidates to investigate strongly non-perturbative theories
even in more complex cases.
