  Direct verification of the existence of an infinite set of multicritical
non-perturbative FPs (Fixed Points) for a single scalar field in two
dimensions, is in practice well outside the capabilities of the present
standard approximate non-perturbative methods. We apply a derivative expansion
of the exact RG (Renormalization Group) equations in a form which allows the
corresponding FP equations to appear as non-linear eigenvalue equations for the
anomalous scaling dimension $\eta$. At zeroth order, only continuum limits
based on critical sine-Gordon models, are accessible. At second order in
derivatives, we perform a general search over all $\eta\ge.02$, finding the
expected first ten FPs, and {\sl only} these. For each of these we verify the
correct relevant qualitative behaviour, and compute critical exponents, and the
dimensions of up to the first ten lowest dimension operators. Depending on the
quantity, our lowest order approximate description agrees with CFT (Conformal
Field Theory) with an accuracy between 0.2\% and 33\%; this requires however
that certain irrelevant operators that are total derivatives in the CFT are
associated with ones that are not total derivatives in the scalar field theory.
