Traditional global search heuristics to solve constraint satisfaction problems focus on properties of an individual variable that mandate early search attention. If, however, one could predict crucial subproblems (the portions of a constraint satisfaction problem likely to cause each other particular difficulty) in advance, search could address them first. This paper postulates several types of crucial subproblems, and shows how local search can be harnessed to identify them before global search for a solution. A variety of heuristics and metrics are then used to guide traditional constraint heuristics with those crucial subproblems. On certain classes of structured problems, such search outperforms traditional heuristics by at least an order of magnitude in both time and space.