We study the statistics of a current flowing through a one-dimensional diffusive channel between a pair of particle reservoirs. Sufficient conditions for a host of dynamical phase transitions both in and out of equilibrium are derived. These transitions manifest themselves as singularities in the large deviation function, which result in enhanced current fluctuations. We present microscopic models implementing each type of these singular behaviors, with possible experimental realizations. Depending on the model, the singularity is associated either with a particle-hole symmetry breaking, which leads to a continuous transition, or with a first-order phase transition in the absence of the symmetry. An exact Landau theory for the different types of transitions is derived.