We report the results of an in-depth analysis of the parameter estimation capabilities of BayesWave, an algorithm for the reconstruction of gravitational-wave signals without reference to a specific signal model. Using binary black hole signals, we compare BayesWave’s performance to the theoretical best achievable performance in three key areas: sky localisation accuracy, signal/noise discrimination, and waveform reconstruction accuracy. BayesWave is most effective for signals that have very compact time-frequency representations. For binaries, where the signal time-frequency volume decreases with mass, we find that BayesWave’s performance reaches or approaches theoretical optimal limits for system masses above approximately 50 Solar masses. For such systems BayesWave is able to localise the source on the sky as well as templated Bayesian analyses that rely on a precise signal model, and it is better than timing- only triangulation in all cases. We also show that the discrimination of signals against glitches and noise closely follow analytical predictions, and that only a small fraction of signals are discarded as glitches at a false alarm rate of 1⁄100 y. Finally, the match between BayesWave-reconstructed signals and injected signals is broadly consistent with first-principles estimates of the maximum possible accuracy, peaking at about 0.95 for high mass systems and decreasing for lower-mass systems. These results demonstrate the potential of unmodelled signal reconstruction techniques for gravitational-wave astronomy.