Context. Coronal waves are an important aspect of the dynamics of the plasma in the corona. Wavefront dislocations are topological features of most waves in nature and also of magnetohydrodynamic waves. Are there dislocations in coronal waves? Aims: The finding and explanation of dislocations may shed light on the nature and characteristics of the propagating waves, their interaction in the corona, and in general on the plasma dynamics. Methods: We positively identify dislocations in coronal waves observed by the Coronal Multi-channel Polarimeter (CoMP) as singularities in the Doppler shifts of emission coronal lines. We study the possible singularities that can be expected in coronal waves and try to reproduce the observed dislocations in terms of localization and frequency of appearance. Results: The observed dislocations can only be explained by the interference of a kink and sausage wave modes propagating with different frequencies along the coronal magnetic field. In the plane transverse to the propagation, the cross-section of the oscillating plasma must be smaller than the spatial resolution, and the two waves result in net longitudinal and transverse velocity components that are mixed through projection onto the line of sight. Alfvén waves can be responsible for the kink mode, but a magnetoacoustic sausage mode is necessary in all cases. Higher (flute) modes are excluded. The kink mode has a pressure amplitude that is less than the pressure amplitude of the sausage mode, though its observed velocity is higher. This concentrates dislocations on the top of the loop. Conclusions: To explain dislocations, any model of coronal waves must include the simultaneous propagation and interference of kink and sausage wave modes of comparable but different frequencies with a sausage wave amplitude much smaller than the kink one. Appendix A is available in electronic form at http://www.aanda.org