  We address the question of determining which mapping class groups of
infinite-type surfaces admit nonelementary continuous actions on hyperbolic
spaces.
  More precisely, let $\Sigma$ be a connected, orientable surface of infinite
type with tame endspace whose mapping class group is generated by a coarsely
bounded subset. We prove that $\mathrm{Map}(\Sigma)$ admits a continuous
nonelementary action on a hyperbolic space if and only if $\Sigma$ contains a
finite-type subsurface which intersects all its homeomorphic translates.
  When $\Sigma$ contains such a nondisplaceable subsurface $K$ of finite type,
the hyperbolic space we build is constructed from the curve graphs of $K$ and
its homeomorphic translates via a construction of Bestvina, Bromberg and
Fujiwara. Our construction has several applications: first, the second bounded
cohomology of $\mathrm{Map}(\Sigma)$ contains an embedded $\ell^1$; second,
using work of Dahmani, Guirardel and Osin, we deduce that
$\mathrm{Map}(\Sigma)$ contains nontrivial normal free subgroups (while it does
not if $\Sigma$ has no nondisplaceable subsurface of finite type), has
uncountably many quotients and is SQ-universal.
