  A subgraph-universal graph/a topological minor-universal graph in a class of
graphs $\mathcal{G}$ is a graph in $\mathcal{G}$ which contains every graph in
$\mathcal{G}$ as a subgraph/topological minor. We prove that the class
$\mathcal{P}$ of all countable planar graphs does not contain a topological
minor-universal graph. This answers a question of Diestel and K\"uhn and
strengthens a result of Pach stating that there is no subgraph-universal graph
in $\mathcal{P}$. Furthermore, we characterise for which subdivided stars $T$
there is a topological minor-universal graph in the class of all countable
$T$-free graphs.
