  We completely classify all neutral or costandard elements in the lattice
$\mathbb{MON}$ of all monoid varieties. Further, we prove that an arbitrary
upper-modular element of $\mathbb{MON}$ except the variety of all monoids is
either a completely regular or a commutative variety. Finally, we verify that
all commutative varieties of monoids are codistributive elements of
$\mathbb{MON}$. Thus, the problems of describing codistributive or
upper-modular elements of $\mathbb{MON}$ are completely reduced to the
completely regular case.
