We prove that for all $\alpha,c>0$ and for all bipartite graphs $H$, all but at most $\alpha n$ vertices of every $cn$-regular graph $G$ whose order $n$ is sufficiently large can be covered by vertex-disjoint copies of $H$. If the vertex classes of $H$ have different size, then even all but a constant number of vertices of $G$ can be covered. This implies that for all $c>0$ and all $r\geq 4$ there exists a constant $C$ such that, in every $cn$-regular graph $G$, all but at most $C$ vertices can be covered by vertex-disjoint subdivisions of $K_r$. We also show that for $r=4,5$ one can take $C=0$.