  Working over imperfect fields, we give a comprehensive classification of
genus-one curves that are regular but not geometrically regular, extending the
known case of geometrically reduced curves. The description is given
intrinsically, in terms of twisted forms of standard models with respect to
infinitesimal group scheme actions, and not via extrinsic equations. The main
new idea is to analyze and exploit moduli for fields of representatives in
Cohen's Structure Theorem. The results serve for the understanding of genus-one
fibrations over higher-dimensional bases.
