Shape segmentation is an active field of research in biomedical imaging. In this context, we present a new parameterization of a snake that is locally refinable. We introduce the possibility of locally increasing the approximation power of the parametric model by inserting basis functions at a specific location. This is controlled by a user-interface that permits the refinement of an initial segmentation around an anchor position selected by a user. Our approach relies on scaling functions that satisfy the refinement relation and are related to wavelets. We also derive explicit formulas for the energy functions associated to our new parameterization. We demonstrate the accuracy of our snake and its robustness under noisy conditions on phantom data. We also present segmentation results on real cell images, which are our main target. The algorithm is made freely available as a plugin for the open source platform Icy.