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 http://mathoverflow.net/questions/71145/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.