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.