In this paper, a class of statistics based on high frequency observations of oscillating and skew Brownian motion is considered. Their convergence rate towards the local time of the underlying process is obtained in form of a functional limit theorem. Oscillating and skew Brownian motion are solutions to stochastic differential equations with singular coefficients: piecewise constant diffusion coefficient or additive local time finite variation term. The result is applied to provide estimators of the skewness parameter and study their asymptotic behavior, and diffusion coefficient estimation is discussed as well. Moreover, in the case of the classical statistics given by the normalized number of crossings, the result is proved to hold for a larger class of Itô processes with singular coefficients. Up to our knowledge, this is the first result proving the convergence rates for estimators of the skewness parameter of skew Brownian motion.