  A character of a group is said to be super-monomial if every primitive
character inducing it is linear. It is conjectured by Isaacs that every
irreducible character of an odd $M$-group is super-monomial. We show that all
non linear irreducible characters of lowest degree of an odd $M$-group is
super-monomial and provide cases in which one can guarantee that certain
irreducible characters of normal subgroups are super-monomial. Finally, we
study groups having two irreducible monomial character degrees.
