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We present a systematic numerical analysis of the magnetic properties of pyramidal-shaped core-shell structures in a size range below 400 nm. These are three-dimensional structures consisting of a ferromagnetic shell which is grown on top of a non-magnetic core. The standard micromagnetic model without the magnetocrystalline anisotropy term is used to describe the properties of the shell. We vary the thickness of the shell between the limiting cases of an ultra-thin shell and a conventional pyramid and delineate different stable magnetic configurations. We find different kinds of single-domain states, which predominantly occur at smaller system sizes. In analogy to equivalent states in thin square films we term these onion, flower, C and S states. At larger system sizes, we also observe two types of vortex states, which we refer to as symmetric and asymmetric vortex states. For a classification of the observed states, we derive a phase diagram that specifies the magnetic ground state as a function of structure size and shell thickness. The transitions between different ground states can be understood qualitatively. We address the issue of metastability by investigating the stability of all occurring configurations for different shell thicknesses. For selected geometries and directions hysteresis measurements are analysed and discussed. We observe that the magnetic behaviour changes distinctively in the limit of ultra-thin shells. The study has been motivated by the recent progress made in the growth of faceted core-shell structures.