  We evaluate the statistical significance of the 3+1 sterile-neutrino
hypothesis using $\nu_e$ and $\bar\nu_e$ disappearance data from reactor, solar
and gallium radioactive source experiments. Concerning the latter, we
investigate the implications of the recent BEST results. For reactor data we
focus on relative measurements independent of flux predictions. For the problem
at hand, the usual $\chi^2$-approximation to hypothesis testing based on Wilks'
theorem has been shown in the literature to be inaccurate. We therefore present
results based on Monte Carlo simulations, and find that this typically reduces
the significance by roughly $1\,\sigma$ with respect to the na\"ive
expectation. We find no significant indication in favor of sterile-neutrino
oscillations from reactor data. On the other hand, gallium data (dominated by
the BEST result) show more than $5\,\sigma$ of evidence supporting the
sterile-neutrino hypothesis, favoring oscillation parameters in agreement with
constraints from reactor data. This explanation is, however, in significant
tension ($\sim 3\,\sigma$) with solar neutrino experiments. In order to assess
the robustness of the signal for gallium experiments we present a discussion of
the impact of cross-section uncertainties on the results.
