Look for situations where two effects are competing, and study which effect "wins," or whether they balance each other in some sense. 

For example, people talk about certain dispersive PDEs being subcritical (dispersion wins over the nonlinear effects) or supercritical (nonlinear effects win). The critical/borderline case is often the most interesting. Other examples arise in combinatorics, where some structure is guaranteed to appear if the size of the problem is sufficiently large, and one may ask where this guarantee begins, i.e. at what size the complexity/unlikeliness of that structure is exactly balanced with the pigeonhole principle.