Confinement of ions in traps is most often motivated by the extended times available for investigations and the possibility to cool the trapped ions' motions nearly to rest, thus avoiding various kinds of disturbances, including Doppler shifts and line broadening. However, specific properties of the confinement itself acting upon the trapped particles may also be used for studies of intrinsic ion properties. Here, we discuss a set of possibilities for high-precision spectroscopy of forbidden transitions in highly charged ions confined in a Penning trap. Such measurements make stringent tests of bound-state QED calculations possible and allow access to fundamental quantities such as the fine structure constant and nuclear properties of the ions.