  We show that given a conformal structure whose holonomy representation fixes
a totally lightlike subspace of arbitrary dimension, there is always a local
metric in the conformal class off a singular set which is Ricci-isotropic and
gives rise to a parallel, totally lightlike distribution on the tangent bundle.
This naturally applies to parallel spin tractors resp. twistor spinors on
conformal spin manifolds and clarifies which twistor spinors are locally
equivalent to parallel spinors. Moreover, we study the zero set of a twistor
spinor using the curved orbit decomposition for parabolic geometries. We can
completely describe its local structure, construct a natural projective
structure on it, and show that locally every twistor spinor with zero is
equivalent to a parallel spinor off the zero set. An application of these
results in low-dimensional split-signatures leads to a complete geometric
description of local geometries admitting non-generic twistor spinors in
signatures (3,2) and (3,3) which complements the well-known description of the
generic case.
