  We consider a prototypical dynamical lattice model, namely, the discrete
nonlinear Schroedinger equation on nonsquare lattice geometries. We present a
systematic classification of the solutions that arise in principal
six-lattice-site and three-lattice-site contours in the form of both discrete
multipole solitons and discrete vortices. Additionally to identifying the
possible states, we analytically track their linear stability both
qualitatively and quantitatively. We find that among the six-site
configurations, the hexapole of alternating phases, as well as the vortex of
topological charge S=2 have intervals of stability; among three-site states,
only the vortex of topological charge S=1 may be stable in the case of focusing
nonlinearity. These conclusions are confirmed both for hexagonal and for
honeycomb lattices by means of detailed numerical bifurcation analysis of the
stationary states from the anticontinuum limit, and by direct simulations to
monitor the dynamical instabilities, when the latter arise. The dynamics reveal
a wealth of nonlinear behavior resulting not only in single-site solitary wave
forms, but also in robust multisite breathing structures.
