  The current most stringent constraints on exotic scalar or tensor couplings
in neutron and nuclear $\beta$ decay, involving left-handed neutrinos, are
obtained from the Fierz interference term. The sensitivity to this term in a
correlation coefficient is usually driven by an energy-averaged kinematic
factor that increases monotonically toward smaller values of the $\beta$
endpoint energies. We first point out here that this property does not hold for
certain differential observables that are directly sensitive to the Fierz term,
such as the $\beta$ or the recoil energy spectrum. This observation is relevant
for the selection of sensitive transitions in searches for exotic couplings
through spectrum shape measurements. We then point out previous errors in the
exploitation of measurements of the $\beta-\nu$ angular correlation coefficient
and discuss their impact on the extraction of constraints on exotic couplings.
