Morphological instabilities in periodic patterns occurring both in precipitation and crystallization processes (Liesegang rings and crystal zoning) are investigated and compared with similar patterns in geological samples (zebra rocks and mud bands in snow sediments). In classical Liesegang systems, undisturbed parallel or concentric precipitation bands are emanated from even or concentric diffusion sources in homogeneous diffusion matrices of gelatine or other gels. In the case of superposing diffusion sources, sources with undulatory curvatures or local diffusion barriers there may occur several types of instabilities within the sequence of regular patterns: (a) gaps within the bands forming radial alleys free of precipitate, (b) transition from broken bands to speckled patterns and (c) apparent branching of bands linked together by so-called anastomoses. Calculations with a competitive particle growth (CPG) model show that lateral instabilities in Liesegang bands (gaps and radial alleys of gaps) are the result of Ostwald ripening effects taking place after precipitation. Apparent branching of bands or formation of anastomoses can be simulated with a prenucleation model according to Ostwald's supersaturation theory. Similar irregularities can be observed in zebra rocks (e.g. banded siderite) whose bandings are commonly explained by sequential sedimentation processes. A very different mechanism is assumed to be responsible for the origin of mud bands in snow sediments. An initially homogeneous distribution of intrinsic mud in snow sediments can be arranged into parallel bands according to a crystal zoning mechanism which is based on repeated thawing and freezing of the snow sediment due to the daily alternation of sun and darkness.