Conflict sets are loci of intersecting wave fronts emanating from $l$ different manifolds. We show that for generic manifolds their conflict sets are projections of smooth conic Lagrangian manifolds. Thus conflict sets locally admit the structure of wave fronts. Simple stable singularities for this problem in $\mathbb{R}^n$ occur when $0\leq n-l \leq 4$. Relations with centre sets, billiards, pedals and orthotomics are also discussed. Throughout canonical relations are used as an essential tool to carry out these geometrical constructions.