  We perform a global analysis of the bounds from charged lepton flavour
violating observables to new physics. We parametrize generic new physics
through the Effective Field Theory formalism and perform global fits beyond the
common one-operator-at-a-time analyses to investigate how much present data is
able to constrain the full parameter space. We particularly focus on leptonic
and semileptonic operators with light quarks, identifying unbounded flat
directions, detailing how many are present and which operators are involved.
The analysis is performed in the general LEFT formalism, which contains all
possible low-energy effective operators relevant for lepton flavour violation,
as well as in more restricted scenarios, when operators come from a SMEFT
completion. We find that flat directions play no role in the fully leptonic
four-fermion operators. Conversely, they significantly hinder the ability to
derive global bounds on semileptonic operators, with several flat or at least
very poorly constrained directions preventing to fully constrain the parameter
space. These results are particularly affected by the proper inclusion of
uncertainties in the parameters describing $\mu-e$ conversion, which decrease
the number of well-constrained directions in operator space when treated as
nuisance parameters in the fit. While present data is able to provide global
constraints on all operators only in the more restricted scenarios we
investigated, very strong correlations among the parameters must exist to avoid
conflict with the different observables. We provide correlation matrices
approximating our full results as a useful tool to compare present data with
particular UV completions.
