This paper is part of an extended study of oxide materials with the mu SR technique. As an example, we present here experimental data on yttria-stabilized zirconia (ZrO2 doped with 8% Y2O3). Three different muon states can be distinguished: i) Deep muonium (less than 17(1)% fraction), seen as a fast-relaxing signal or indirectly via decoupling measurements in high longitudinal fields, ii) mu(+) in a paramagnetic environment 62(6)% fraction), characterized by a very weak but clearly-visible hyperfine interaction, and iii) diamagnetic muon 21(1)% fraction); the diamagnetic signal is broadened only by the interaction with nuclear moments. The state corresponding to mu(+) in a paramagnetic environment and the diamagnetic state are attributed to the same (oxygen-bound) muon configuration, but we assume that they have different electron surroundings (with or without an unpaired electron in the vicinity). The paramagnetic electron is not captured in the Coulomb potential of the positive muon but is self-trapped (polaron formation) at a nearby Zr ion. The distant electron interacts with the muon only via dipolar magnetic fields. This explains the very weak hyperfine interaction felt by the mu(+) state in a paramagnetic environment. A further result of the experiment is that the disappearance of this signal with increasing temperature is not due to ionization of an electron shallowly bound to the muon but is caused by rapid spin fluctuations of the electron, averaging the hyperfine interaction to zero.