Hallin and Ley [Bernoulli 18 (2012) 747–763] investigate and fully characterize the Fisher singularity phenomenon in univariate and multivariate families of skew-symmetric distributions. This paper proposes a refined analysis of the (univariate) problem, showing that singularity can be more or less severe, inducing n 1/4 (“simple singularity”), n 1/6 (“double singularity”), or n 1/8 (“triple singularity”) consistency rates for the skewness parameter. We show, however, that simple singularity (yielding n 1/4 consistency rates), if any singularity at all, is the rule, in the sense that double and triple singularities are possible for generalized skew-normal families only. We also show that higher-order singularities, leading to worse-than-n 1/8 rates, cannot occur. Depending on the degree of the singularity, our analysis also suggests a simple reparametrization that offers an alternative to the so-called centred parametrization proposed, in the particular case of skew-normal and skew-t families, by Azzalini [Scand. J. Stat. 12 (1985) 171–178], Arellano-Valle and Azzalini [J. Multivariate Anal. 113 (2013) 73–90], and DiCiccio and Monti [Quaderni di Statistica 13 (2011) 1–21], respectively.