  Boundary integral equations and Nystrom discretization provide a powerful
tool for the solution of Laplace and Helmholtz boundary value problems.
However, often a weakly-singular kernel arises, in which case specialized
quadratures that modify the matrix entries near the diagonal are needed to
reach a high accuracy. We describe the construction of four different
quadratures which handle logarithmically-singular kernels. Only smooth
boundaries are considered, but some of the techniques extend straightforwardly
to the case of corners. Three are modifications of the global periodic
trapezoid rule, due to Kapur-Rokhlin, to Alpert, and to Kress. The fourth is a
modification to a quadrature based on Gauss-Legendre panels due to
Kolm-Rokhlin; this formulation allows adaptivity. We compare in numerical
experiments the convergence of the four schemes in various settings, including
low- and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace
problems. We also find striking differences in performance in an iterative
setting. We summarize the relative advantages of the schemes.
