  The experimental sensitivity to $\mu \to e$ conversion on nuclei is set to
improve by four orders of magnitude in coming years. However, various operator
coefficients add coherently in the amplitude for $\mu \to e$ conversion,
weighted by nucleus-dependent functions, and therefore in the event of a
detection, identifying the relevant new physics scenarios could be difficult.
Using a representation of the nuclear targets as vectors in coefficient space,
whose components are the weighting functions, we quantify the expectation that
different nuclear targets could give different constraints.We show that all but
two combinations of the 10 Spin-Independent (SI) coefficients could be
constrained by future measurements, but discriminating among the axial, tensor
and pseudoscalar operators that contribute to the Spin-Dependent (SD) process
would require dedicated nuclear calculations. We anticipate that $\mu \to e$
conversion could constrain 10 to 14 combinations of coefficients; if $\mu \to e
\gamma$ and $\mu \to 3e$ constrain eight more, that leaves 60 to 64 "flat
directions" in the basis of QED$\times$QCD-invariant operators which describe
$\mu \to e$ flavour change below $m_W$.
