  Saito's microlocalization construction has been used to great effect in
understanding hypersurface singularities. In this paper, we introduce what we
believe to be a suitable analogue of the microlocalization construction for
local complete intersection subvarieties. As evidence, we relate our
construction to Saito's in the codimension one case. Moreover, we use this
construction to study various natural questions concerning the minimal exponent
of LCI subvarieties. We show that the minimal exponent agrees with the smallest
Bernstein-Sato root, which was expected to be true. We also show that, in the
isolated complete intersection singularities case, the minimal exponent agrees
with the smallest non-zero spectral number, as defined by Dimca, Maisonobe and
Saito. As applications of these results, we prove constructibility of the
minimal exponent along certain Whitney stratifications and we prove that the
spectrum (hence, the minimal exponent) is constant in equisingular families of
ICIS varieties, in the sense of Gaffney.
