The quadratic assignment problem (QAP) is a very difficult and
practically relevant combinatorial optimization problem which has
attracted much research effort. Local search (LS) moves can be quickly
evaluated on the QAP, and hence favoured methods tend to be hybrids of
global optimization schemes and LS.  Here we introduce the 
{\em multiobjective} QAP (mQAP) where  $m \geq 2$ distinct QAPs must be
minimized simultaneuously over the same permutation space, and hence
we require a set of solutions approximating the Pareto front (PF).  We
argue that the best way to organise a hybrid LS for the mQAP will
depend on details of the multiobjective fitness landscape.  By using
various techniques and measures to probe the landscapes of mQAPs, we
attempt to find evidence for the relative ease with which the
following can be done by LS: approach the PF from a random initial
solution, or search along or close to the PF itself. On the basis of
such explorations, we hope to design an appropriate hybrid LS for this
problem. The paper contributes a number of landscape measurement
methods that we believe are generally appropriate for multiobjective
combinatorial optimization.