  Understanding instabilities in dynamical systems drives to the heart of
modern chaos theory, whether forecasting or attempting to control future
outcomes. Instabilities in the sense of locally maximal stretching in maps is
well understood, and is connected to the concepts of Lyapunov
exponents/vectors, Oseledec spaces and the Cauchy--Green tensor. In this paper,
we extend the concept to global optimization of stretching, as this forms a
skeleton organizing the general instabilities. The `map' is general but
incorporates the inevitability of finite-time as in any realistic application:
it can be defined via a finite sequence of discrete maps, or a finite-time flow
associated with a continuous dynamical system. Limiting attention to
two-dimensions, we formulate the global optimization problem as one over a
restricted class of foliations, and establish the foliations which both
maximize and minimize global stretching. A classification of nondegenerate
singularities of the foliations is obtained. Numerical issues in computing
optimal foliations are examined, in particular insights into special curves
along which foliations appear to veer and/or do not cross, and foliation
behavior near singularities. Illustrations and validations of the results to
the H\'{e}non map, the double-gyre flow and the standard map are provided.
