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

Semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities