We present a survey of 41 Kepler Objects of Interest (KOIs) for exomoons using Bayesian photodynamics, more than tripling the number of KOIs surveyed with this technique. We find no compelling evidence for exomoons although 13 KOIs yield spurious detections driven by instrumental artifacts, stellar activity, and/or perturbations from unseen bodies. Regarding the latter, we find seven KOIs exhibiting >5 sigma evidence of transit timing variations, including the "mega-Earth" Kepler-10c, likely indicating an additional planet in that system. We exploit the moderately large sample of 57 unique KOIs surveyed to date to infer several useful statistics. For example, although there is a diverse range in sensitivities, we find that we are sensitive to Pluto-Charon mass-ratio systems for Asymptotically = to40% of KOIs studied and Earth-Moon mass-ratios for 1 in 8 cases. In terms of absolute mass, our limits probe down to 1.7 Ganymede masses, with a sensitivity to Earth-mass moons for 1 in 3 cases studied and to the smallest moons capable of sustaining an Earth-like atmosphere (0.3 M) for 1 in 4. Despite the lack of positive detections to date, we caution against drawing conclusions yet, since our most interesting objects remain under analysis. Finally, we point out that had we searched for the photometric transit signals of exomoons alone, rather than using photodynamics, we estimate that 1 in 4 KOIs would have erroneously been concluded to harbor exomoons due to residual time correlated noise in the Kepler data, posing a serious problem for alternative methods.