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.
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