We formulate a resolution of singularities algorithm for analyzing the zero
sets of real-analytic functions in dimensions $\geq 3$. Rather than using the
celebrated result of Hironaka, the algorithm is modeled on a more explicit and
elementary approach used in the contemporary algebraic geometry literature. As
an application, we compute the critical integrability index for real-analytic
functions and obtain the sharp growth rate of their sublevel sets. This also
leads to a characterization of the oscillation index of scalar oscillatory
integrals with real-analytic phases in all dimensions.