A Majorana neutrino is characterized by just one flavor diagonal electromagnetic form factor, the anapole moment, which in the static limit corresponds to the axial vector charge radius $〈{r}_{A}^{2}〉.$ Experimental information on this quantity is scarce, especially in the case of the tau neutrino. We present a comprehensive analysis of the available data on the single photon production process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\nu}\overline{\ensuremath{\nu}}\ensuremath{\gamma}$ off Z resonance, and we discuss the constraints that these measurements can set on $〈{r}_{A}^{2}〉$ for the $\ensuremath{\tau}$ neutrino. We also derive limits for the Dirac case, when the presence of a vector charge radius $〈{r}_{V}^{2}〉$ is allowed. Finally, we comment on additional experimental data on ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ scattering from the NuTeV, E734, CCFR, and CHARM-II Collaborations, and estimate the limits implied for $〈{r}_{A}^{2}〉$ and $〈{r}_{V}^{2}〉$ for the muon neutrino.