The core structures of hedgehog point disclinations in Heisenberg magnets and Lebwohl-Lasher nematic liquid crystals have been investigated by Monte Carlo simulation. We find qualitative agreement with the theoretical predictions of Schopohl and Sluckin that the magnetic defects have smaller cores. We also find some evidence for the picture of Penzen-stadler and Trebin in which spherical symmetry of the nematic hedgehog core is broken, and the core region can be considered as a disclination line of index 1/2. Finally we make some general comments about the significance of our results in the context of other recent theoretical speculations