  We establish what semi-discrete linear Weingarten surfaces with
Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian
spaceforms are, confirming their required properties regarding curvatures and
parallel surfaces, and then classify them. We then define and analyze their
singularities. In particular, we discuss singularities of (1) semi-discrete
surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of
semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean
curvature $1$ surfaces in de Sitter $3$-space. We include comparisons with
different previously known definitions of such singularities.
