  We classify regularity for a class of Lagrangian mean curvature type
equations, which includes the potential equation for prescribed Lagrangian mean
curvature and those for Lagrangian mean curvature flow self-shrinkers and
expanders, translating solitons, and rotating solitons. We first show that
convex viscosity solutions are regular provided the Lagrangian angle or phase
is $C^2$ and convex in the gradient variable. We next show that for merely
H\"older continuous phases, convex solutions are regular if they are
$C^{1,\beta}$ for sufficiently large $\beta$. Singular solutions are given to
show that each condition is optimal and that the H\"older exponent is sharp.
Along the way, we generalize the constant rank theorem of Bian and Guan to
include arbitrary dependence on the Legendre transform.
