For a magnetic remanence to be useful to paleomagnetists, it must be stable. Single domain (SD) grains generally have high stability because the magnetization can only rotate uniformly. In larger grains, the magnetization is divided into domains and the magnetization can change by movement of domain walls or transitions between different numbers of domains. It is often assumed that stability in non-SD grains is controlled mainly by pinning forces that prevent the domain walls from moving. However, transitions should be equally important. In this article, a three-dimensional numerical micromagnetic model is used to study the critical points at which transitions occur. Hysteresis curves (including minor branches) are calculated for cuboids with no internal stress or magnetocrystalline anisotropy. There are two main kinds of transition: turning points and pitchfork bifurcations. At a turning point, the susceptibility goes to infinity and there is a jump in the magnetization. Most jumps in hysteresis loops are turning points. At a pitchfork bifurcation, for example at the Curie point, there is a jump in susceptibility but the magnetization changes continuously. It is shown that the transition from an SD to a non-SD state is a pitchfork bifurcation, the generalization of curling mode nucleation in spheroids. In a hysteresis loop, there is often a large gap in field between nucleation and the first irreversible jump in magnetization. This gap is similar to a gap that is often seen experimentally between the formation of a small spike domain and the appearance of a full size body domain. Since large changes in magnetization can occur reversibly in small grains with low internal stress, they may be more resistant to AF demagnetization than grains with high internal stress. However, they are not necessarily more resistant to thermal demagnetization.