Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye. Tanveer, S. and Speziale, Charles G. Unspecified Center NAS1-18605; NAS1-19480; DE-FG02-92ER-14270; NSF DMS-91-07608; RTOP 505-90-52-01...