An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable polynomial splits in $K$, but no separable polynomial splits in $K$?
Splitting fields can be taken with respect to any family of polynomials over a fixed field $K$. Fields $E$ that are splitting fields of families of polynomials over a field $K$ are precisely those fields such that $E/K$ is normal. Galois extensions are defined to be normal and separable. If the family is finite, say $f_1,\ldots,f_s$, then $E$ is the splitting field of $f=f_1\cdots f_s$, and $E/K$ (which is always normal) will be separable if and only if the irreducible factors of $f$ are separable.