The dynamics of line singularities in three different physical systems are considered, namely vortices in an inviscid fluid, vortices in a type-II supercondctor, and dislocations in an elastic crystal. When the core of the singularity can be regularised with a continuum model, as is the case for superconducting and fluid vortices, the dynamics can be derived systematically in the asymptotic limit as the core radius tends to zero. The asymptotic analysis is more difficult when the core of the singularity is so small as to demand an atomic model, as is the case for dislocations, where the derivation of a law of motion is still an open problem in the mathematical sense.