  Stanislaw Ulam asked whether there exists a universal countable planar graph
(that is, a countable planar graph that contains every countable planar graph
as a subgraph). J\'anos Pach (1981) answered this question in the negative. We
strengthen this result by showing that every countable graph that contains all
countable planar graphs must contain (i) an infinite complete graph as a minor,
and (ii) a subdivision of the complete graph $K_t$ with multiplicity $t$, for
every finite $t$.
  On the other hand, we construct a countable graph that contains all countable
planar graphs and has several key properties such as linear colouring numbers,
linear expansion, and every finite $n$-vertex subgraph has a balanced separator
of size $O(\sqrt{n})$. The graph is $\mathcal{T}_6\boxtimes P_{\!\infty}$,
where $\mathcal{T}_k$ is the universal treewidth-$k$ countable graph (which we
define explicitly), $P_{\!\infty}$ is the 1-way infinite path, and $\boxtimes$
denotes the strong product. More generally, for every positive integer $t$ we
construct a countable graph that contains every countable $K_t$-minor-free
graph and has the above key properties.
  Our final contribution is a construction of a countable graph that contains
every countable $K_t$-minor-free graph as an induced subgraph, has linear
colouring numbers and linear expansion, and contains no subdivision of the
countably infinite complete graph (implying (ii) above is best possible).
