Let $X$ be a reduced scheme, so, generically regular; you may assume extra conditions like equidimensional and seminormal (though normal is stronger than I'd like, as is Gorenstein). 

> Is there a reasonable definition of "section of the anticanonical bundle" over $X$? At the _very_ least I'd like to be able to restrict such a "section" to the regular locus and get an actual section. If $X$ is normal, then I'd like any (honest) anticanonical section over $X_{reg}$ to uniquely extend to one of these objects on $X$.

The motivation comes from Frobenius splitting. If $X = X_{reg}$ is defined over a perfect field of characteristic $p$, and $\sigma$ is a section of the anticanonical over $X$ (not vanishing on any component), then we can define a map $\varphi:\ F_* \mathcal O_X \to  \mathcal O_X$ by $g \mapsto \mathcal C(g/\sigma) \sigma$ away from $\sigma=0$, which extends over all of $X$ (here $F$ is the Frobenius and $\mathcal C$ is the Cartier operator on (top) forms). Such maps make sense even when $X$ is singular, but I'd rather talk about an anticanonical section than about $\varphi$.

If $X$ is Gorenstein, then I suppose an "anticanonical section" should be a section of the dual of the canonical line bundle, but my $X$s usually aren't Gorenstein. 

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