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We show that given a conformal structure whose holonomy representation fixesa totally lightlike subspace of arbitrary dimension, there is always a localmetric in the conformal class off a singular set which is Ricci-isotropic andgives rise to a parallel, totally lightlike distribution on the tangent bundle.This naturally applies to parallel spin tractors resp. twistor spinors onconformal spin manifolds and clarifies which twistor spinors are locallyequivalent to parallel spinors. Moreover, we study the zero set of a twistorspinor using the curved orbit decomposition for parabolic geometries. We cancompletely describe its local structure, construct a natural projectivestructure on it, and show that locally every twistor spinor with zero isequivalent to a parallel spinor off the zero set. An application of theseresults in low-dimensional split-signatures leads to a complete geometricdescription of local geometries admitting non-generic twistor spinors insignatures (3,2) and (3,3) which complements the well-known description of thegeneric case.

Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors