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The purpose of this paper is to revive interest in the problem of caustics with symmetry. Small imperfections in the point focus of a perfect lens are classified by the Seidel aberrations, while catastrophe theory deals satisfactorily with large ones. But the middle ground where an initial wave front is not perfectly spherical, and yet retains some symmetry, has proved more difficult to occupy. Caustics from water-drop lenses provide a motivation, but the analysis given is not restricted to such lenses. We start from a water-drop lens whose periphery has n-fold rotational symmetry combined with reflection symmetry (Dn). In 3D the unperturbed caustic is a flared and scalloped double cone with n rib lines. When n is even, the two cones are mutually rotated. Perturbing the drop by one of the terms suggested by catastrophe theory produces a sequence of caustics. It unfolds the original caustic. With parabolic umbilic singularities acting as off-stage organising centres, butterfly singularities and beak-to-beak events change the number of cusps in a cross-section until the caustic may eventually become n − 2 cusped triangles. We present such sequences for a number of low-order perturbations and find that identical sequences are produced if the perturbation is held fixed and the screen is moved. The two ways are related by power laws. For n = 6, a practical way of making a perturbation would be provided by the Bragg bubble model of a metal. Iluminating a single bubble in one of the hexagonally symmetric rafts would produce a hexagonally symmetric caustic that could be perturbed by distorting the raft elastically. The treatment can be extended to cover the chiral point groups Cn. [ABSTRACT FROM AUTHOR]