  We present a systematic survey of scarring and symmetry effects in the
stadium billiard. The localization of individual eigenfunctions in Husimi phase
space is studied first, and it is demonstrated that on average there is more
localization than can be accounted for on the basis of random-matrix theory,
even after removal of bouncing-ball states and visible scars. A major point of
the paper is that symmetry considerations, including parity and time-reversal
symmetries, enter to influence the total amount of localization. The properties
of the local density of states spectrum are also investigated, as a function of
phase space location. Aside from the bouncing-ball region of phase space,
excess localization of the spectrum is found on short periodic orbits and along
certain symmetry-related lines; the origin of all these sources of localization
is discussed quantitatively and comparison is made with analytical predictions.
Scarring is observed to be present in all the energy ranges considered. In
light of these results the excess localization in individual eigenstates is
interpreted as being primarily due to symmetry effects; another source of
excess localization, scarring by multiple unstable periodic orbits, is smaller
by a factor of $\sqrt{\hbar}$.
