We investigate the dependence of geometric properties of Radon measures, such as Hausdorff dimension and rectifiability of singular sets, on the wavefront set. We prove our results by adapting the method of Brummelhuis to the non-analytic case. As an application we obtain a general form of the Uncertainty Principle for measures on the complex sphere. In particular we construct the spectral decomposition of spherical harmonics (on a three dimensional sphere it is even a spectral basis) which allows us to generalize the celebrated theorem of Aleksandrov and Forelli concerning regularity of pluriharmonic measures. Joint work with Rami Ayoush.