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We analyze polarization singularities of optical beams that propagate through crystals possessing structural dislocations. We show that screw dislocations of crystalline structure can lead to the appearance of purely screw-type dislocations of light wavefronts. This can happen only in crystals that belong to trigonal and cubic systems. These polarization singularities will give rise to optical vortices with the topological charge equal to ±1, whenever a crystal sample is placed between crossed circular polarizers. We have also found that edge dislocations present in the cubic and trigonal crystals, with the Burgers vector perpendicular to the three-fold symmetry axes, can impose mixed screw-edge dislocations in the wavefronts of optical beams and generate singly charged optical vortices. The results of our analysis can be applied for detecting and identifying dislocations of different types available in crystals. [ABSTRACT FROM AUTHOR]