  In this paper, we deal with the problem of monogenity of number fields
defined by monic irreducible trinomials $F(x)=x^{12}+ax^m+b\in \mathbb{Z}[x]$
with $1\leq m\leq11$. We give sufficient conditions on $a$, $b$, and $m$ so
that the number field $K$ is not monogenic. In particular, for $m=1$ and for
every rational prime $p$, we characterize when $p$ divides the index of $K$ and
we provide a partial answer to the Problem $22$ of Narkiewicz \cite{Nar} for
these number fields. Our results are illustrated by computational examples.
