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. A graph is critically apex if the deletion of every single vertex yields a planar graph. Similarly (critically) edge-apex graphs are defined with the deletion of edges. Furthermore, k-apex and k-edge-apex graphs are defined by deleting k vertices or edges. The data set provides graphs in Graph6 data format.