Given a graph, layout problems are concerned with choosing an ordering of its vertices so that adjacent vertices lie close together in the ordering. For example, the aim might be to minimize the sum over all edges of the separation of the edge's endpoints in the ordering. We investigate problems of this type for graphs generated randomly, by percolation or by continuum analogs. Such graphs have been used by computer scientists for comparing heuristics for these problems. If time permits we may also discuss some connectivity properties of these graphs.