Let $k$ be a field. If $f \in k[x]$ is a polynomial, and $d/dx\ f = 0$, then either

$f$ is constant, or

$char\ k = p$ and $f \in k[x^p]$.

So "annihilated by all derivations" is perhaps not the right thing to ask for in characteristic $p$ (though that's what I asked for in Is the singular locus ideal preserved by all derivations? ).

What is the right thing to ask for?

I would like an invariance condition one could state of a subscheme $Y$ of $X$, that holds for the singular locus, but doesn't hold for (say) any regular closed point on a rational variety.