  We consider here the existence and structure of trapped surfaces, horizons
and singularities in spherically symmetric static massless scalar field
spacetimes. Earlier studies have shown that there exists no event horizon in
such spacetimes if the scalar field is asymptotically flat. We extend this
result here to show that this is true in general for spherically symmetric
static massless scalar field spacetimes, whether the scalar field is
asymptotically flat or not. Other general properties and certain important
features of these models are also discussed.
