Murata and Umehara gave a classification of complete flat fronts in the Euclidean 3-space and proved their orientability. Here, a flat front is a flat surface (i.e., a surface where one of the principal curvatures is identically zero) with admissible singularities. In this paper, we investigate wave fronts where one of the principal curvatures is a non-zero constant. Although they are orientable in the regular surface case, there exist non-orientable examples. We classify weakly complete ones and derive their orientability.