An \emph{$H$-packing} in a graph $G$ is a collection of pairwise vertex-disjoint copies of $H$ in $G$. We prove that for every $c > 0$ and every bipartite graph $H$, any $\lfloor cn \rfloor$-regular graph $G$ admits an $H$-packing that covers all but a constant number of vertices. This resolves a problem posed by Kühn and Osthus in 2005. Moreover, our result is essentially tight: the conclusion fails if $G$ is not both regular and sufficiently dense, it is in general not possible to guarantee covering all vertices of $G$ by an $H$-packing, and if $H$ is non-bipartite then $G$ need not contain any copies of $H$. We also prove that for all $c > 0$, integers $t \geq 2$, and sufficiently large $n$, all the vertices of every $\lfloor cn \rfloor$-regular graph can be covered by vertex-disjoint subdivisions of $K_t$. This resolves another problem of Kühn and Osthus from 2005, which goes back to a conjecture of Verstraëte from 2002. Our proofs combine novel methods for balancing expanders and super-regular subgraphs with a number of powerful techniques including properties of robust expanders, regularity lemma, and blow-up lemma.