  Using some adaptations of the adjacency decomposition method \cite{CR} and
the program {\it cdd} (~\cite{Fu}), we compute the first computationally
difficult cases of convex cones of $m$-ary and oriented analogs of semi-metrics
and cut semi-metrics, which were introduced in \cite{DR2} and \cite{DP}. We
considered also more general notion of $(m,s)$-super-metric and corresponding
cones. The data on related cones - the number of facets, of extreme rays, of
their orbits and diameters - are collected in Table \ref{tab:MainLovelyTable}.
We study also criterion of adjacency for skeletons of those cones and their
duals. Some families of extreme rays and operations on them are also given.
