  The support poset of a monomial ideal $I\subseteq\mathbf{k}[x_1,\dots,x_n]$
encodes the relation between the variables $x_1,\dots,x_n$ and the minimal
monomial generators of $I$. It is known that not every poset is realizable as
the support poset of some monomial ideal. We describe some posets $P$ for which
we can explicitly find at least one monomial ideal $I_P$ such that $P$ is the
support poset of $I_P$. Also, for some families of monomial ideals we describe
their support posets and study their properties. As an example of application
we examine the relation between forests and series-parallel ideals.
