  We explore new ways of regulating defect behavior in systems of conservation
laws. Contrary to usual regularization schemes (such as a vanishing viscosity
limit), which attempt to control defects by making them smoother, our schemes
result in defects which are \textit{more singular}, and we thus refer to such
schemes as "irregularizations". In particular, we seek to produce \textit{delta
shock} defects which satisfy a condition of \textit{stationarity}. We are
motivated to pursue such exotic defects by a physical example arising from
dislocation dynamics in materials physics, which we describe.
