  Noncommutative geometry has seen remarkable applications for high energy
physics, viz. the geometrical interpretation of the Standard Model. The
question whether it also allows for supersymmetric theories has so far not been
answered in a conclusive way. In this first of three papers we do a systematic
analysis of the possibilities for almost-commutative geometries on a
4-dimensional, flat background to exhibit not only a particle content that is
eligible for supersymmetry but also have a supersymmetric action. We come up
with an approach in which we identify the basic 'building blocks' of
potentially supersymmetric theories and the demands for their action to be
supersymmetric. Examples that satisfy these demands turn out to be sparse.
