Motivation: The generic freeness lemma tells us that, given a finite-type morphism $f \colon X \to Y$ of affine noetherian integral schemes, there is a dense open subset $U \subset Y$ such that $f^{-1}(U) \to U$ is flat and surjective. In particular, all the fibers over $U$ necessarily have the same dimension and are ``equivalent up to flat deformation." So, in some sense, a computable version of generic freeness would allow us to classify all the fibers of a morphism (and not just, say, the general fiber).
Such a computable version is described in Vasconcelos' 1997 paper "Flatness testing and torsionfree morphisms," Theorem 2.1, although Vasconcelos gives the impression that this "computable version" was already well-known in certain circles. If we are looking at a ring homomorphism$$A \longrightarrow B = A[T_1,\dotsc,T_n]/I,$$the only computationally nontrivial part of the algorithm is to compute a Gröbner basis for $I$, in the sense described in this question.
[Qualification 1: If you actually want to compute a stratification, you also need to be able to find the irreducible components of $A/(f)$ so that the next step will have an integral base. This is not computationally trivial, but if $A$ is of finite type over a field, it does have standard implementations, for instance, in Macaulay2.]
[Qualification 2: Judging by the next remark in Vasconcelos' paper, it may be possible to get by if you can compute a Gröbner basis for $I$ over the fraction field of $A$. Macaulay2 can do this--I think--but according to the documentation, it is not remotely efficient.]
