Moire effects ('Glass patterns') that occur in the superposition of correlated random layers have long been studied. However, only little is known of Glass patterns which occur in the 1D random case: namely, when the superposed layers consist of random line gratings, straight or curved. In this paper we study the properties and the behaviour of such Glass patterns, and we compare them with those of the analogous moire effects between periodic line gratings. We provide for each case a detailed mathematical analysis of the fringe shapes and locations, along with illustrative figures which clearly compare the behaviour of the corresponding random and periodic (or repetitive) superpositions