  Three static and axially symmetric (Weyl-type) ring singularities -- the
Majumdar-Papapetrou--type (extremally charged) ring, the Bach-Weyl ring and the
Appell ring -- are studied in general relativity in order to show how
remarkably the geometries in their vicinity differ from each other. This is
demonstrated on basic measures of the rings and on invariant characteristics
given by the metric and by its first and second derivatives (lapse,
gravitational acceleration and curvature), and also on geodesic motion. The
results are also compared against the Kerr space-time which possesses a ring
singularity too. The Kerr solution is only stationary, not static, but in spite
of the consequent complication by dragging, its ring appears to be simpler than
the static rings. We show that this mainly applies to the Bach-Weyl ring,
although this straightforward counter-part of the Newtonian homogeneous
circular ring is by default being taken as the simplest ring solution, and
although the other two static ring sources may seem more "artificial". The
weird, directional deformation around the Bach-Weyl ring probably indicates
that a more adequate coordinate representation and interpretation of this
source should exists.
