The nonpolarized points (NPP) of the Q and U Stokes parameters of the CMB can be classified according to the geometry of the polarization field. We describe a procedure to identify these points in the pixelized sky and present the shape of the polarization angles in the vicinity of NPPs. We design a test of Gaussianity using the Kullback-Leibler divergence. We show that the total number density of nonpolarized points of the E and B families is closely related to the presence of lensing and the tensor-to-scalar ratio r. We further show that in the absence of lensing, the total number of NPPs of all types does not depend on r, while the lensing effect removes this degeneracy. This analysis is applied to the CMB maps from the 2018 Planck release. We show that there is a general consistency of SMICA and NILC maps compared to a reference set of Gaussian simulations. The strongest discrepancies are found in the Commander (with corresponding p value 0.07) and SEVEM (p = 0.01) maps.