  We prove that any finitely generated torsion free solvable subgroup of the
group ${\rm IET}$ of all Interval Exchange Transformations is virtually
abelian.
  In contrast, the lamplighter groups $A\wr \mathbb{Z}^k$ embed in ${\rm IET}$
for every finite abelian group $A$, and we construct uncountably many non
pairwise isomorphic 3-step solvable subgroups of ${\rm IET}$ as semi-direct
products of a lamplighter group with an abelian group.
  We also prove that for every non-abelian finite group $F$, the group $F\wr
\mathbb{Z}^k$ does not embed in ${\rm IET}$.
