  We construct the ${\cal N}=1$ supersymmetric extension of Double Field Theory
for Riemannian and the non-Riemannian in a unified approach. The inclusion of
fermions in the double geometry force us to use the generalized frame formalism
to construct the generalized flux components for these geometries. We focus on
the most general prescription required to get the $D=10$ minimal supergravity
model. We study how to consistently avoid the gauge fixing procedure of the
double Lorentz symmetry when $n=\bar n$ and $0\le n \le 5$, which gives rise to
a bigravity structure (pair of vielbeins producing the same non-Riemannian
degrees of freedom). As an example we show how to to include fermionic degrees
of freedom in the type I torsional Newton-Cartan (TNC) theory ($n=\bar n=1$)
which is related to Carrollian geometries and stringy Newton Cartan through
duality rotations and/or null reductions/uplifts.
