Many local geometric structures, such as group actions, Poisson structures, symplectic reduced spaces, singularity theory, ... , rely on using those vector fields that respect said structure to give local normal forms and consequent local product structures. In this talk I will describe how one can create a general approach to such questions, making vector fields the principal ingredient. This leads naturally to foliations (or stratifications) on the one hand and singular spaces on the other. It is then natural to combine the two... This is joint work with Mark Roberts (Surrey).