  In this article we start a systematic study of the bi-Lipschitz geometry of
lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly
embed into Hamming cubes with distortion at most~$6$. It follows that
lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We
study the metric behaviour of the operation of taking the lamplighter graph
over the vertex-coalescence of two graphs. Based on this analysis, we provide
metric characterizations of superreflexivity in terms of lamplighter graphs
over star graphs or rose graphs. Finally, we show that the presence of a clique
in a graph implies the presence of a Hamming cube in the lamplighter graph over
it. An application is a characterization in terms of a sequence of graphs with
uniformly bounded degree of the notion of trivial Bourgain-Milman-Wolfson type
for arbitrary metric spaces, similar to Ostrovskii's characterization
previously obtained in \cite{ostrovskii:11}.
