  Fold singular points play important roles in the theory of maximal surfaces.
For example, if a maximal surface admits fold singular points, it can be
extended to a timelike minimal surface analytically. Moreover, there is a
duality between conelike singular points and folds. In this paper, we
investigate fold singular points on spacelike surfaces with non-zero constant
mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces
do not admit fold singular points. Moreover, we show that the singular point
set of any conjugate CMC surface of a spacelike Delaunay surface with conelike
singular points consists of (2,5)-cuspidal edges.
