Density of the smooth locus

If $X\subseteq {\mathbb{A}}^n$ is irreducible of dimension $d$, choose a nonzero $(n-d)$-rowed minor of a Jacobian matrix at one point of minimum tangent dimension. Its nonvanishing locus is a nonempty Zariski-open set on which the Jacobian has rank $n-d$, so every point there is smooth. Irreducibility makes every nonempty Zariski-open set dense.

For an affine variety over a perfect field, the tangent-space codimension at a point is the rank of the Jacobian matrix of defining equations. The smooth locus of a pure $d$-dimensional variety in ${\mathbb{A}}^N$ is where this rank is $N-d$.