Our study of Goursat distributions originates new types of k-contact distributions and Lie systems with applications. In particular, families of generators for Goursat distributions on \(\mathbb {R}^4, \mathbb {R}^5\) and \(\mathbb {R}^6\) give rise to Lie systems and we characterise Goursat structures that are k-contact distributions. Our results are used to study the zero-trailer and other systems via Lie systems and k-contact manifolds. New ideas for the development of superposition rules via geometric structures and the characterisation of k-contact distributions are given and applied. Some relations of k-contact geometry with parabolic Cartan geometries are inspected.