In this data set we provide the list of non-planar connected graphs with different apex properties. We call a graph apex if there is a vertex whose deletion gives a planar graph. Independence is considered in the 3-dimensional generic rigidity matroid. A graph is (3,6)-sparse if if for every subset of n vertices, with at least 3 elements, the number of edges in the subgraph induced by these elements is at most 3n − 6. The data set provides graphs in Graph6 data format.