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This study investigates Bayesian signature-domain inference of hydrological models using Approximate Bayesian Computation (ABC) algorithms, and compares it to "traditional" time-domain inference. Our focus is on the quantification of predictive uncertainty in the streamflow time series and on understanding the information content of particular combinations of signatures. A combination of synthetic and real data experiments using conceptual rainfall-runoff models is employed. Synthetic experiments demonstrate: (i) the general consistency of signature and time-domain inferences, (ii) the ability to estimate streamflow error model parameters (reliably quantify streamflow uncertainty) even when calibrating in the signature domain, and (iii) the potential robustness of signature-domain inference when the (probabilistic) hydrological model is misspecified (e.g., by unaccounted timing errors). The experiments also suggest limitations of the signature-domain approach in terms of information loss when general (nonsufficient) statistics are used, and increased computational costs incurred by the ABC implementation. Real data experiments confirm the viability of Bayesian signature-domain inference and its general consistency with time-domain inference in terms of predictive uncertainty quantification. In addition, we demonstrate the utility of the flashiness index for the estimation of streamflow error parameters, and show that signatures based on the Flow Duration Curve alone are insufficient to calibrate parameters controlling streamflow dynamics. Overall, the study further establishes signature-domain inference (implemented using ABC) as a promising method for comparing the information content of hydrological signatures, for prediction under data-scarce conditions, and, under certain circumstances, for mitigating the impact of deficiencies in the formulation of the predictive model.