  The aim of this paper is to study supersoluble skew braces, a class of skew
braces that encompasses all finite skew braces of square-free order. It turns
out that finite supersoluble skew braces have Sylow towers, and that in an
arbitrary supersoluble skew brace $B$ many relevant skew brace-theoretical
properties are easier to identify: for example, a centrally nilpotent ideal of
$B$ is $B$-centrally nilpotent, a fact that simplifies the computational search
for the Fitting ideal; also, $B$ has finite multipermutational level if and
only if $(B,+)$ is nilpotent.
  Given a finite presentation of the structure skew brace $G(X,r)$ associated
with a finite non-degenerate solution of the Yang--Baxter Equation (YBE), there
is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover,
supersoluble skew braces are examples of almost polycyclic skew braces, so they
give rise to solutions of the YBE on which one can algorithmically work on.
