To my knowledge, graph isomorphism for graphs with bounded clique width or bounded rank width is open.
2015 arxiv paper claims it is polynomial:
We give an algorithm that, for every fixed $k$, decides isomorphism of graphs of rank width at most $k$ in polynomial time. As the clique width of a graph is bounded in terms of its rank width, we also obtain a polynomial time isomorphism test for graph classes of bounded clique width.
The paper is 48 pages and doesn't appear published in peer reviewed journal. There is only one citation on Scholar.google.
Q1 Is graph isomorphism still open for bounded clique width or bounded rank width?