We present a novel method of moving snake curves over 3D meshes. Our snake is made from the vertices of a triangular mesh, so it never leaves the 3D surface, eliminating any need for mapping on to 2D and subsequent remapping on to the 3D mesh. Our snake will find features related to Gaussian curvature, or ridges and valleys. We move each vertex of the snake to one of its neighbor vertices on the mesh, while reducing the energy of the snake, which expresses distance from a feature, by means of a greedy optimization. This is fast enough to support real-time interaction. We handle changes of snake topology using 3D versions of methods originally developed for images. These can handle collisions within the snake, and hence split the snake and find multiple features. We demonstrate the performance of our algorithm on several example meshes, including human faces.