  For any infinite-type surface $S$, a natural question is whether the homology
of its mapping class group contains any non-trivial classes that are supported
on (i) a compact subsurface or (ii) a finite-type subsurface. Our purpose here
is to study this question, in particular giving an almost-complete answer when
the genus of $S$ is positive (including infinite) and a partial answer when the
genus of $S$ is zero. Our methods involve the notion of shiftable subsurfaces
as well as homological stability for mapping class groups of finite-type
surfaces.
