  The magnetic properties of single-domain nanoparticles with different
geometric shapes, crystalline anisotropies and lattice structures are
investigated. A recently proposed scaling approach is shown to be universal and
in agreement with dimensional analysis coupled with an assumption of {\em
incomplete} self-similarity. It is used to obtain phase diagrams of magnetic
nanoparticles featuring three competing configurations: in-plane and
out-of-plane ferromagnetism and vortex formation. The influence of the vortex
core on the scaling behavior and phase diagram is analyzed. Three-dimensional
phase diagrams are obtained for cylindrical nanorings, depending on their
height, outer and inner radius. The triple points in these phase diagrams are
shown to be in linear relationship with the inner radius of the ring.
Elliptically shaped magnetic nanoparticles are also studied. A new
parametrization for double vortex configurations is proposed, and regions in
the phase diagram are identified where the double vortex is a stable ground
state.
