  In this paper we construct singular solutions to the critical Dirac equation
on spheres. More precisely, first we construct solutions admitting two points
singularities that we call Delaunay-type solutions because of their
similarities with the Delaunay solutions constructed for the singular Yamabe
problem in \cite{MP1 , Schoen1989}. Then we construct another kind of singular
solutions admitting a great circle as a singular set. These solutions are the
building blocks for singular solutions on a general Spin manifold.
