  A new characterization of the singular packing subspaces of general bounded
self-adjoint operators is presented, which is used to show that the set of
operators whose spectral measures have upper packing dimension equal to one is
a $G_\delta$ (in suitable metric spaces). As an application, it is proven that,
generically (in space of continuous sampling functions), spectral measures of
the limit-periodic Schr\"odinger operators have upper packing dimensions equal
to one. Consequently, in a generic set, these operators are quasiballistic.
