Complements of discriminants of real singularities of type $X_{10}$

A conjecturally complete list of connected components of complements of discriminant varieties (aka wave fronts) of smooth function singularities of type $X_{10}^3$ and $X_{10}^1$ is presented; it are the first examples of not semi-quasihomogeneous plane function singularities. It is shown that the complements of discriminants of singularities of classes $X_9^{\pm }$ and $X_{10}^1$ have non-trivial 1-dimensional homology groups, in contrast to all simple singularity classes.