  In this paper we investigate the singularities of Lagrangian mean curvature
flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I
singularities can only occur at certain times determined by invariants in the
cohomology of the initial data. In the type II case, these smooth singularity
models are asymptotic to special Lagrangian cones; hence all type II
singularities are modeled by unions of special Lagrangian cones.
