  We show that all static spacetimes in higher dimensions are of Weyl types G,
I_i, D or O. This applies also to stationary spacetimes if additional
conditions are fulfilled, as for most known black hole/ring solutions. (The
conclusions change when the Killing generator becomes null, such as at Killing
horizons.) Next we demonstrate that the same Weyl types characterize warped
product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas
warped spacetimes with a two-dimensional Lorentzian factor are restricted to
the types D or O. By exploring the Bianchi identities, we then analyze the
simplest non-trivial case from the above classes - type D vacuum spacetimes,
possibly with a cosmological constant, dropping, however, the assumptions that
the spacetime is static, stationary or warped. It is shown that for ``generic''
type D vacuum spacetimes the corresponding principal null directions are
geodetic in any dimension (this applies also to type II spacetimes). For n>=5,
however, there may exist particular cases of type D spacetimes which admit
non-geodetic multiple principal null directions and we present such examples in
any n>=7. Further studies are restricted to five dimensions, where the type D
Weyl tensor is described by a 3x3 matrix \Phi_{ij}. In the case with
``twistfree'' (A_{ij}=0) principal null geodesics we show that in a ``generic''
case \Phi_{ij} is symmetric and eigenvectors of \Phi_{ij} coincide with those
of the expansion matrix S_{ij}, providing us with three preferred spacelike
directions of the spacetime. Similar results are also obtained when relaxing
the twistfree condition and assuming instead that \Phi_{ij} is symmetric. The
n=5 Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are
briefly studied as specific examples of type D vacuum spacetime.
