  We explore the rigidity of generic frameworks in 3-dimensions whose
underlying graph is close to being planar. Specifically we consider apex
graphs, edge-apex graphs and their variants and prove independence results in
the generic 3-dimensional rigidity matroid adding to the short list of graph
classes for which 3-dimensional rigidity is understood. We then analyse global
rigidity for these graph classes and use our results to deduce bounds on the
maximum likelihood threshold of graphs in these nearly planar classes.
