Let $G$ and $H$ be two graphs, each one of them being a path, a cycle or a star. In this paper, we determine the $b$-chromatic number of every subdivision-vertex neighbourhood corona $G\boxdot H$ or $G\boxdot K_n$, where $K_n$ is the complete graph of order $n$. It is also established for those graphs $K_n\boxdot G$ having $m$-degree not greater than $n+2$. All the proofs are accompanied by illustrative examples.