The paper contains a theoretical investigation of those multivalued singular solutions to the shallow shell equations which correspond physically to concentrated loads and dislocations. Use of the shell equations in complex form permits a unified treatment of the load and dislocation problems. The analysis is limited to the case of shells with a quadratic middle surface, and Fourier transforms of the solutions are obtained. Complete solutions in closed form in the case of a shallow sphere are given in the Appendix, including some results not previously published.