The polarization singularities in vector wavefields have been extensively studied analytically and experimentally. The polarization singularities can be analyzed by using electromagnetic theory or Stokes parameters, or be described in terms of complex Stokes scalar fields. In some practical applications, partially coherent beams have more advantages than fully coherent beams. Recently, the concept of the polarization singularities has been extended from fully coherent beams to partially coherent beams. In this paper, using the representation of cross-spectral density matrix propagation, the explicit propagation expressions for the partially coherent edge dislocation beams are derived in free space, and based on the spectral Stokes parameters the spectral singularities are studied in detail. It is shown that there exist spectral s12, s23 and s31 singularities of partially coherent edge dislocation beams in free-space propagation. s12 singularities correspond to circular polarization (C-points) of the partially coherent edge dislocation beams, and s30 (s30) means right-(left-) handedness, where the orientations of the major and minor axes of the polarization ellipse become undefined. s23 and s31 singularities must be located on L-lines, where the handedness of the polarization ellipse is undetermined (linear polarization). The motion, creation and annihilation of spectral Stokes singularities may appear in the variation of a controlling parameter, such as off-axis distance, slope of edge dislocation, spatial correlation length, or in the variation of the propagation distance. By suitably varying the spatial correlation length or propagation distance the V-point, the handedness reversal of C-point, creation and annihilation for a pair of oppositely charged spectral singularities take place. The creation and annihilation occur for a pair of s12 singularities with opposite topological charge but same handedness. The critical points of the controlling parameters and propagation distance, at which pairs of different spectral singularities annihilate, are not the same. The collision of the C-point and L-line results in a V-point (vector singularity), which is unstable. A small perturbation leads to the handedness reversal. At such a point the state of polarization is undetermined and the degree of polarization P=0. The results obtained in this paper would be useful for a deep understanding of polarization singularities of stochastic electromagnetic beams.