The flashback of the investigation of CuGeO3 doped with magnetic impurities carried out by high-frequency EPR brings to light physics still actual for one-dimensional S = 1/2 quantum spin chains and covering a vast area from disorder-driven quantum critical phenomena to a new type of magnetic oscillations. It is shown that a key opening the door for a better understanding of this field of research is the Oshikawa and Affleck (OA) theory and, especially, following from it the universal link between the line width and g factor. The most puzzling problem appears around the staggered field, which contributes to different physical properties and serves as a driving force in the onset of anomalous growth of the line width and g factor at low temperatures. Accent is made on unsolved problems still providing a challenge for the theoretical explanation, including the genesis of the staggered field in doped systems, the contribution of the staggered magnetization to the integrated intensity, and spin susceptibility of a Griffiths phase and its magnetic properties on the nanoscale. A new type of magnetic oscillations, having explanation neither in OA theory nor within the framework of the semi-classical magnetization motion, is described in detail. This experimental finding poses the most difficult case and a touchstone for theory, as long as the corresponding modes of magnetic oscillations may be likely treated as violating Landau–Lifshits equation of motion.