The performance of a minimum-B Electron Cyclotron Resonance Ion Source (ECRIS) is traditionally quantified by measuring the beam current and quality of the extracted ion beams of different charge state ions. The stability of the extracted ion beam currents has drawn more attention recently as the technology is pushing its limits toward higher ion charge states and beam intensities. The stability of the extracted beam is often compromised by plasma instabilities manifesting themselves as rapid oscillations of the beam currents in millisecond scale. This paper focuses on practical aspects of diagnostics techniques of the instabilities, showcases examples of instability-related diagnostics signals, and links them to the plasma physics of ECR ion sources. The reviewed techniques include time-resolved microwave emission diagnostics, bremsstrahlung measurements, direct measurement of electron and ion fluxes, measurement of the ion beam energy spread, and optical emission diagnostics. We list the advantages and disadvantages of each technique and outline the development needs of further diagnostics. Finally, we discuss the implications of the instabilities in both historical and forward-looking context of ECRIS development.