Sharp corners and edges are ubiquitous in modem technology and found on the tip of an atomic force microscope (AFM) and the vertices and edges of crystalline materials. The microhydrodynamics of particles with sharp corners and edges are special in that geometric discontinuities give rise to traction singularities for viscous flow around the particles. Such traction fields are analyzed with both analytical and numerical solution of a second kind integral equation. The traction field in the vicinity of a vertex has a singular structure that is more intricate than simple superposition of the well-known edge singularities. The new insights into the structure of the vertex singularities are used to formulate and test spectral element strategies for numerical solution of viscous flow past sharp corners and edges, The best schemes constructed to date show remarkable agreement with the analytical results.