Features characterizing the perfection of high-quality YNi2B2C, HoNi2B2C and TbNi2B2C single crystals grown by the floating zone technique with optical heating in the TSFZ regime are investigated. The grown crystals are crack free, single phase and single crystalline,in the whole cross-section. Various origins of small fractions of minority phases in several crystals impurities, phase diagram features and limits of growth parameters are revealed. For the crystals a subgrain misorientation < 0.15 degrees was determined by neutron diffraction. The YNi2B2C crystals show an extremely narrow superconducting transition width 0.1 K and high residual resistivity ratios up to 48. Longitudinal and radial element segregation phenomena with particular respect to the light elements are related to the various modifications of superconducting properties and T-c variations in HoNi2B2C single crystals. The interface shape and radial segregation suggest a noticeable convection flow in HoNi2B2C melts. (c) 2004 Elsevier B.V. All rights reserved.