Miller's paper constructs two examples. The first was discovered by Banchoff and Strauss in 1979. The polyhedron is show in the top left figure below.
Several triangles are coplanar, in groups of four, and of two.
The second example constructed by Miller is new. Several pairs of adjacent triangles are coplanar. It remains open to make a complete list of all icosamonohedra, as well as to classify the combinatorial types of all monohedra.