We show that second-generation gravitational-wave detectors at their design sensitivity will allow us to directly probe the ringdown phase of binary black hole coalescences. This opens the possibility to test the so-called black hole no-hair conjecture in a statistically rigorous way. Using state-of-the-art numerical relativity-tuned waveform models and dedicated methods to effectively isolate the quasistationary perturbative regime where a ringdown description is valid, we demonstrate the capability of measuring the physical parameters of the remnant black hole and subsequently determining parameterized deviations from the ringdown of Kerr black holes. By combining information from O(5) binary black hole mergers with realistic signal-to-no...