  We describe the topological behavior of the conjugacy action of the mapping
class group of an orientable infinite-type surface $\Sigma$ on itself. Our main
results are:
  (1) All conjugacy classes of $MCG(\Sigma)$ are meager for every $\Sigma$,
  (2) $MCG(\Sigma)$ has a somewhere dense conjugacy class if and only if
$\Sigma$ has at most two maximal ends and no non-displaceable finite-type
subsurfaces,
  (3) $MCG(\Sigma)$ has a dense conjugacy class if and only if $\Sigma$ has a
unique maximal end and no non-displaceable finite-type subsurfaces.
  Our techniques are based on model-theoretic methods developed by Kechris,
Rosendal and Truss.
