A Goursat structure on a manifold of dimension n is a rank two distributionƊ such that dim Ɗ (i) = i + 2, for 0 ≤ i ≤ n-2,where Ɗ (i) denote the elements of the derived flag ofƊ, defined by Ɗ (0) = Ɗ and Ɗ (i+1) = Ɗ (i) + [Ɗ (i),Ɗ (i)].Goursat structures appeared first in the work ofvon Weber and Cartan,who have shown that on an open and dense subset they can be converted into theso-called Goursat normal form. Later, Goursat structures have been studied byKumpera and Ruiz. In the paper, we introduce a new local invariant for Goursatstructures, called the singularity type, and prove that the growth vector andthe abnormal curves of all elements of the derived flag are determined by thisinvariant. We provide a detailed analysis of all abnormal and rigid curves ofGoursat structures. We show that neither abnormal curves, if n ≥ 6, norabnormal curves of all elements of the derived flag, if n ≥ 9, determinethe local equivalence class of a Goursat structure. The latter observation isdeduced from a generalized version of Bäcklund's theorem. We also propose anew proof of a classical theorem of Kumpera and Ruiz. All results areillustrated by the n-trailer system, which, as we show, turns out to be auniversal model for all local Goursat structures.