The principles of Weighing Designs, their connection with general linear statistical hypothesis model, and the limitations of weighing designs, when the design matrix is singular, have been discussed in full length. This Dissertation is mainly study of the structure of the singular weighing designs with reference to the questions of precision of estimable linear functions of the weights vis-a-vis their precision obtained from the corresponding best design matrices under the non-singular set-up. Inter alia, some theorems on the invariance properties of estimable linear fucntions of the weights of the objects have been proven in the sequel. Finally, an optimum augmentation procedure of the singular basis in the frame work of generalized inverse of matrices.