Plasticity, a key property in the mechanical behavior and processing of crystalline solids, has been traditionally viewed as a smooth and homogeneous flow. However, using two experimental methods, acoustic emission and high-resolution extensometry, to probe the collective dislocation dynamics in various single crystals, we show that its intermittent critical-like character appears as a rule rather than an exception. Such intermittent, apparently scale-free plastic activity is observed in single-slip as well as multislip conditions and is not significantly influenced by forest hardening. Strain bursts resulting from dislocation avalanches are limited in size by a nontrivial finite size effect resulting from the lamellar character of avalanches. This cutoff explains why strain curves of macroscopic samples are smooth, whereas fluctuations of plastic activity are outstanding in submillimetric structures.