  In this paper we study solutions to elliptic linear equations
$L(u)=\partial_i(a^{ij}(x)\partial_j u) + b^i(x) \partial_i u + c(x) u=0$,
either on $R^n$ or a Riemannian manifold, under the assumption of Lipschitz
control on the coefficients $a^{ij}$. We focus our attention on the critical
set $Cr(u)\equiv\{x:|\nabla u|=0\}$ and the singular set
$S(u)\equiv\{x:u=|\nabla u|=0\}$, and more importantly on effective versions of
these. Currently, under the coefficient control we have assumed, the strongest
results in the literature say that the singular set is n-2-dimensional, however
at this point it has not even been shown that $H^{n-2}(S)<\infty$ unless the
coefficients are smooth. Fundamentally, this is due to the need of an
$\epsilon$-regularity theorem which requires higher coefficient control as the
frequency increases. We introduce new techniques for estimating the critical
and singular set, which avoids the need of any such $\epsilon$-regularity.
Consequently, we prove that if the frequency of u is bounded by $\Lambda$ then
we have the estimates $H^{n-2}(C(u))\leq C^{\Lambda^2}$, $H^{n-2}(S(u))\leq
C^{\Lambda^2}$, depending on whether the equation is critical or not. More
importantly, we prove corresponding estimates for the {\it effective} critical
and singular sets. Even under the assumption of analytic coefficients these
results are much sharper than those currently in the literature. We also give
applications of the technique to the nodal set of solutions, and to give
estimates on the corresponding eigenvalue problem.
