Quasicrystalline as well as crystalline faceted single grains of four phases were obtained during directional crystallisation of an Al-Cu-Fe alloy by the Bridgman technique. The monoclinic phase, Al13(Cu, Fe)4, dominating at high temperatures formed single-crystal lamellae 0.5 mm to 1 mm thick. A second type of attractive morphological form exhibiting flux dissolution terraces was observed on spherical single crystals of phase Al(Fe, Cu). Rectangular, hexagonal and octagonal shaped dissolution terraces were revealed at the positions of two-, three- and four-fold symmetry axes, respectively. A single quasicrystalline phase, Al6Cu2Fe, exhibited icosahedral symmetry with growth forms of a dodecahedron with pentagonal facets. The flux dissolution of the phase apparently plays an essential role in a peritectic reaction leading to quasicrystalline phase formation. Polygonal single grains of phase Al7Cu2Fe exhibiting tetragonal symmetry formed the fourth type of thermodynamically stable growth forms. Single grains of the phase crystallised in the form of pellets with an octagonal cross-section. The growth morphology of the stable phases was investigated by scanning electron microscopy. The chemical composition of the growth forms described was confirmed by X-ray microanalysis using a scanning electron microscope, whereas the phase composition was determined using electron selected area diffraction and X-ray powder diffraction.