  In this paper we define and analyze singularities of discrete linear
Weingarten surfaces with Weierstrass-type representations in $3$-dimensional
Riemannian and Lorentzian spaceforms. In particular, we discuss singularities
of discrete surfaces with non-zero constant Gaussian curvature, and parallel
surfaces of discrete minimal and maximal surfaces, and discrete constant mean
curvature $1$ surfaces in de Sitter $3$-space, including comparisons with
different previously known definitions of such singularities.
