The talk is concerned with the structure of the module of derivations and its connections with singularities and vector fields of varieties. Modules of derivations are not well understood -- despite great advances on the Zariski-Lipman conjecture, there is still no complete characterization for when they are free. Our work is partially motivated by a question of Poincaré, who asked how to decide whether a polynomial vector field in the complex plane leaves some algebraic curve invariant. We reformulate this problem in terms of bounding from below the initial degree of the module consisting of all vector fields that leave a fixed curve invariant. This module is a quotient of the module of derivations. We also treat the case of smooth varieties of higher dimension. For plane curves of low genus we establish a correspondence between finer invariants of the module of derivations, such as its graded betti numbers, and properties of the singularities of the curve. This is joint work with Chardin, Hassanzadeh, Simis, and Ulrich.