In the theory of partial coherence, coherence singularities can occur in the spectral degree of coherence (SDOC): in case the fields at two different points are completely uncorrelated, the phase of the SDOC is undefined. For a partially coherent vortex beam, the detection of coherence singularities is linked to the measurement of topological charge, whose magnitude equals the number of ring dislocations in its far field amplitude. However, the phase distribution of coherence singularities is rarely mentioned in the literature and the amplitude distribution can hardly reflect the sign of topological charge. In this letter, we present a phase-analysis method for measuring the coherence singularities by introducing a movable perturbation at a certain point in an illumination window of a finite size. Using the proposed method, we measure experimentally the coherence singularities of a partially coherent vortex beam in the focal plane. From the results, the magnitude and sign of the topological charge can be determined simultaneously from the phase distribution of the coherence singularities. Our results can find application in information transmission.