  We study mean curvature flow of Lagrangians in $\mathbb{C}^n$ that are
cohomogeneity-one with respect to a compact Lie group $G \leq \mathrm{SU}(n)$
acting linearly on $\mathbb{C}^n$. Each such Lagrangian necessarily lies in a
level set $\mu^{-1}(\xi)$ of the standard moment map $\mu \colon \mathbb{C}^n
\to \mathfrak{g}^*$, and mean curvature flow preserves this containment. We
classify all cohomogeneity-one self-similarly shrinking, expanding and
translating solutions to the flow, as well as cohomogeneity-one smooth special
Lagrangians lying in $\mu^{-1}(0)$. Restricting to the case of
almost-calibrated flows in the zero level set $\mu^{-1}(0)$, we classify
finite-time singularities, explicitly describing the Type I and Type II blowup
models. Finally, given any cohomogeneity-one special Lagrangian in
$\mu^{-1}(0)$, we show it occurs as the Type II blowup model of a Lagrangian
MCF singularity. Throughout, we give explicit examples of suitable group
actions, including a complete list in the case of $G$ simple. This yields
infinitely many new examples of shrinking and expanding solitons for Lagrangian
MCF, as well as infinitely many new singularity models.
