A congruence is a surface in the Grassmannian Gr(1,P3)of lines in pro-jective 3-space. To a space curveC, we associate the Chow hypersurface in Gr(1,P3)consisting of all lines which intersectC. We compute the singular locus of this hy-persurface, which contains the congruence of all secants toC. A surfaceSinP3defines the Hurwitz hypersurface in Gr(1,P3)of all lines which are tangent toS. Weshow that its singular locus has two components for general enoughS: the congru-ence of bitangents and the congruence of inflectional tangents. We give new proofsfor the bidegrees of the secant, bitangent and inflectional congruences, using ge-ometric techniques such as duality, polar loci and projections. We also study thesingularities of these congruences.