We mainly investigate the polarization singularities of the focused azimuthally polarized (AP) beams modulated by spiral phase and sector obstacles. The results reveal that either the spiral phase or sector obstacle can convert the central V-point to C-points, C-point dipoles, or even double V-points under certain conditions. The conversion can be selectively controlled by appropriately setting the topological charge of the spiral phase and the sector angle of the obstacle. These results may have implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused cylindrically polarized beams.