Let $X = \operatorname{Spec} k[x_1,\ldots,x_n]/I$ be an affine variety of dimension $d$. By definition the singular locus of $X$ is the locus of points of $X$ where the rank of the Jacobian matrix is $< n - d$.

In theory the singular locus is easy to compute: its ideal is generated by $I$ and all the $(n - d) \times (n - d)$ minors of the Jacobian.

In practice, computing the singular locus can be infeasible. For instance, suppose $X \subset \mathbf{A}^{12}$ is a dimension $4$ variety defined by $18$ equations. (Let us assume this is a minimal number of equations for $X$.) The the Jacobian is an $18 \times 12$ matrix, and one must compute all the $8 \times 8$ minors of this matrix, of which there are over $21$ million! Even a computer will struggle greatly with this task.

Are there any known algorithms or techniques for getting a handle on the singularities of $X$ when faced with such a computationally difficult situation?