  Some coherent sheaves on projective varieties have a non reduced versal
deformation space. For example, this is the case for most unstable rank 2
vector bundles on ${\mathbb P}_2$. In particular, it may happen that some
moduli spaces of stable sheaves are non reduced.
  We consider the case of some sheaves on ribbons (double structures on smooth
projective curves): the quasi locally free sheaves of rigid type. Le $E$ be
such a sheaf.
  -- Let ${\mathcal E}$ be a flat family of sheaves containing $E$. We find
that it is a reduced deformation of $E$ when some canonical family associated
to ${\mathcal E}$ is also flat.
  -- We consider a deformation of the ribbon to reduced projective curves with
two components, and find that $E$ can be deformed in two distinct ways to
sheaves on the reduced curves. In particular some components $\bf M$ of the
moduli spaces of stable sheaves deform to two components of the moduli spaces
of sheaves on the reduced curves, and $\bf M$ appears as the "limit" of
varieties with two components, whence the non reduced structure of $\bf M$.
