  We present a classification of the possible quantum deformations of the
supergroup $GL(1|1)$ and its Lie superalgebra $gl(1|1)$. In each case, the
(super)commutation relations and the Hopf structures are explicitly computed.
For each $R$ matrix, one finds two inequivalent coproducts whether one chooses
an unbraided or a braided framework while the corresponding structures are
isomorphic as algebras. In the braided case, one recovers the classical algebra
$gl(1|1)$ for suitable limits of the deformation parameters but this is no
longer true in the unbraided case.
