  A graph $G$ is \emph{locally irregular} if no two of its adjacent vertices
have the same degree. In [Fioravantes et al. Complexity of finding maximum
locally irregular induced subgraph. {\it SWAT}, 2022], the authors introduced
and studied the problem of finding a locally irregular induced subgraph of a
given a graph $G$ of maximum order, or, equivalently, computing a subset $S$ of
$V(G)$ of minimum order, whose deletion from $G$ results in a locally irregular
graph; $S$ is denoted as an \emph{optimal vertex-irregulator of $G$}. In this
work we provide an in-depth analysis of the parameterised complexity of
computing an optimal vertex-irregulator of a given graph $G$. Moreover, we
introduce and study a variation of this problem, where $S$ is a substet of the
edges of $G$; in this case, $S$ is denoted as an \emph{optimal edge-irregulator
of $G$}. In particular, we prove that computing an optimal vertex-irregulator
of a graph $G$ is in FPT when parameterised by the vertex integrity,
neighborhood diversity or cluster deletion number of $G$, while it is
$W[1]$-hard when parameterised by the feedback vertex set number or the
treedepth of $G$. In the case of computing an optimal edge-irregulator of a
graph $G$, we prove that this problem is in FPT when parameterised by the
vertex integrity of $G$, while it is NP-hard even if $G$ is a planar bipartite
graph of maximum degree $4$, and $W[1]$-hard when parameterised by the size of
the solution, the feedback vertex set or the treedepth of $G$. Our results
paint a comprehensive picture of the tractability of both problems studied
here, considering most of the standard graph-structural parameters.
