  Neutrino physics is nowadays receiving more and more attention as a possible
source of information for the long-standing problem of new physics beyond the
Standard Model. The recent measurement of the mixing angle $\theta_{13}$ in the
standard mixing oscillation scenario encourages us to pursue the still missing
results on leptonic CP violation and absolute neutrino masses. However,
puzzling measurements exist that deserve an exhaustive evaluation. The NESSiE
Collaboration has been setup to undertake conclusive experiments to clarify the
muon-neutrino disappearance measurements at small $L/E$, which will be able to
put severe constraints to models with more than the three-standard neutrinos,
or even to robustly measure the presence of a new kind of neutrino oscillation
for the first time. To this aim the use of the current FNAL-Booster neutrino
beam for a Short-Baseline experiment has been carefully evaluated. This
proposal refers to the use of magnetic spectrometers at two different sites,
Near and Far. Their positions have been extensively studied, together with the
possible performances of two OPERA-like spectrometers. The proposal is
constrained by availability of existing hardware and a time-schedule compatible
with the CERN project for a new more performant neutrino beam, which will
nicely extend the physics results achievable at the Booster. The possible FNAL
experiment will allow to clarify the current $\nu_{\mu}$ disappearance tension
with $\nu_e$ appearance and disappearance at the eV mass scale. Instead, a new
CERN neutrino beam would allow a further span in the parameter space together
with a refined control of systematics and, more relevant, the measurement of
the antineutrino sector, by upgrading the spectrometer with detectors currently
under R&D study.
