We study the polarization structures in the vicinity of C-lines in the near fields diffracted from a pair of small holes. We find that, when the incident light is circularly polarized, both the true C-lines and the structures near them are controlled only by the longitudinal component. Furthermore, we find that all the existing singular lines of circular polarization have the winding number ±1, which is very different than the usual numbers ±1/2, and the structure of major axes of the polarization ellipses surrounding these lines are shown to form structures different than the Möbius strip type. All these features prove to be stable upon small changes of shapes or positions of the apertures. However, C-lines with a unit winding number split into two C-lines of half-winding numbers when the incident light is elliptically polarized light.