  A congruence is a surface in the Grassmannian $\mathrm{Gr}(1,\mathbb{P}^3)$
of lines in projective $3$-space. To a space curve $C$, we associate the Chow
hypersurface in $\mathrm{Gr}(1,\mathbb{P}^3)$ consisting of all lines which
intersect $C$. We compute the singular locus of this hypersurface, which
contains the congruence of all secants to $C$. A surface $S$ in $\mathbb{P}^3$
defines the Hurwitz hypersurface in $\mathrm{Gr}(1,\mathbb{P}^3)$ of all lines
which are tangent to $S$. We show that its singular locus has two components
for general enough $S$: the congruence of bitangents and the congruence of
inflectional tangents. We give new proofs for the bidegrees of the secant,
bitangent and inflectional congruences, using geometric techniques such as
duality, polar loci and projections. We also study the singularities of these
congruences.
