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Edgecritical graphs
A graph is said to be edgecritical if its chromatic number drops whenever an edge is deleted. Every edgecritical graph is also vertexcritical. The following collection was calculated by Gordon Royle (source) and lists edgecritical graphs on 4 up to 12 vertices.

Fullerenes
A fullerene is a cubic planar graph having all faces 5 or 6cycles. Fullerenes are planar and polyhedral, and every fullerene has exactly twelve 5cycles. The following collection was calculated by the House of Graphs (source) and list all fullerenes on up to 90 vertices.

Maximal trianglefree graphs
A trianglefree graph is an undirected connected graph in which no three vertices form a triangle of edges. Trianglefree graphs may be equivalently defined as graphs with clique number 2, graphs with girth 4 (or 0), graphs with no induced 3cycle, or locally independent graphs.
In order to... 
Planar graphs
A planar graph is a graph that can be embedded in the plane, i.e. it can be drawn on the plane in such a way that its edges intersect only at their endpoints. The following collection was calculated by Brendan McKay (source) and list all nonisomorphic connected planar graphs on up to 9 vertices....

Vertexcritical graphs
A graph is said to be vertexcritical if its chromatic number drops whenever a vertex is deleted. The following collection was calculated by Gordon Royle (source) and lists vertexcritical graphs on 4 up to 11 vertices.