Near the sample edge, or a sharp magnetic field step, the drift of two-dimensional (2D) electrons in a magnetic field has the form of skipping and snake orbits. We show that families of skipping and snake orbits of electrons injected at one point inside a 2D metal generically exhibit caustics folds, cusps, and cusp triplets, and, in one exceptional case, an extreme section of the butterfly bifurcation. Periodic appearance of singularities along the +/- B interface leads to the magneto-oscillations of nonlocal conductance in multiterminal electronic devices.