The aim of the talk is to highlight some features of the theory of non-singular webs in the geometry of volume-preserving transformations. The local structure of these "divergence-free webs" is far richer than that of their classical counterparts due to the presence of an ambient volume form which interacts with the web. This is reflected in the existence of several local invariants which can be non-trivial even for webs which are parallelizable. They range from curvature invariants derived from the canonical connection of a divergence-free web (which was first defined by S. Tabachnikov in his work on Lagrangian and Legendrian 2-webs) to purely geometric ones inspired by the results of W. Blaschke, G. Bol and G. Thomsen on planar 3-web holonomy. We will construct and characterize these invariants, show how their triviality relates to the triviality of the corresponding divergence-free web, and discuss potential applications of the underlying theory in numerical relativity. Joint work with Wojciech Domitrz.