Spontaneous radiation emitted from relativistic electrons undergoing betatron motion in a plasma-focusing channel is analyzed, and applications to plasma wake-field accelerator experiments and to the ion-channel laser (ICL) are discussed. Important similarities and differences between a free electron laser (FEL) and an ICL are delineated. It is shown that the frequency of spontaneous radiation is a strong function of the betatron strength parameter a(beta), which plays a role similar to that of the wiggler strength parameter in a conventional FEL. For a(beta) > or approximately 1, radiation is emitted in numerous harmonics. Furthermore, a(beta) is proportional to the amplitude of the betatron orbit, which varies for every electron in the beam. The radiation spectrum emitted from an electron beam is calculated by averaging the single-electron spectrum over the electron distribution. This leads to a frequency broadening of the radiation spectrum, which places serious limits on the possibility of realizing an ICL.