  In this paper, we describe on the one hand, all relative minimal models Y of
a minuscule Schubert variety X using some combinatorics on quivers and we prove
that the morphism from Y to X is small (in the sense of intersection
cohomology).
  On the other hand, thanks to a result of B. Totaro, any small resolution Z of
a minuscule Schubert variety X has to be a relative minimal model of X. So X
admits a small resolution if and only if there exists a smooth relative minimal
model Y of X. We give a combinatoric criterion for Y to be smooth describing in
this way all small resolutions of X.
  We also use stringy polynomials and the relative canonical model to give
another way to tell when a minuscule Schubert variety admits a small
resolution.
