  For each integer $n\ge 2$, we identify new infinite families of monogenic
trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which
have small Galois group. These families are thus different from many previous
examinations of specific trinomial forms in the literature. Moreover, in
certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce
asymptotics on the number of such trinomials with $A\le X$.
