Helimagnets realize an effective lamellar ordering that supports disclination and dislocation defects. Here, we investigate the micromagnetic structure of screw dislocation lines in cubic chiral magnets using analytical and numerical methods. The far field of these dislocations is universal and classified by an integer strength ν that quantifies its Burgers vector. We demonstrate that a rich variety of dislocation-core structures can be realized even for the same strength ν. In particular, the magnetization at the core can be either smooth or singular. We present a specific example with ν=1 for which the core is composed of a chain of singular Bloch points. In general, screw dislocations carry a noninteger but finite skyrmion charge so that they can be efficiently manipulated by spin currents and should contribute to the topological Hall effect.