In this paper elementary boundary integral equations for the Helmholtz equation in the exterior domain, based on Green's formula or through representation of the solution by layer potentials, are considered. Even when the partial differential equation has a unique solution, for any given closed boundary I, these elementary boundary integral equations can be shown to be singular at a countable set of characteristic wavenumbers. Spectral properties and conditioning of the boundary integral operators and their discrete boundary element counterparts are studied near characteristic wavenumbers, with a view to assessing the suitability of these formulations for the solution of the exterior Helmholtz equation. Collocation methods are used for the discretisation of the boundary integral equations which are either of the Fredholm first kind, second kind, or hyper-singular type. The effect of quadrature errors on the accuracy of the discrete collocation methods is systematically investigated

Revised software for the BEM solution of Helmholtz problems is now available from the author's BEMHELM package.