jats:titleSummary</jats:title>jats:pIn this paper, we use an enriched approximation space for the efficient and accurate solution of the Helmholtz equation to solve problems of wave scattering by polygonal obstacles. This is implemented in both boundary element method (BEM) and partition of unity BEM (PUBEM) settings. The enrichment draws upon the asymptotic singular behaviour of scattered fields at sharp corners, leading to a choice of fractional‐order Bessel functions that complement the existing Lagrangian (BEM) or plane wave (PUBEM) approximation spaces. Numerical examples consider configurations of scattering objects, subject to the Neumann “sound hard” boundary conditions, demonstrating that the approach is a suitable choice for both convex scatterers and also for multiple scattering objects that give rise to multiple reflections. Substantial improvements are observed, significantly reducing the number of degrees of freedom required to achieve a prescribed accuracy in the vicinity of a sharp corner.</jats:p>