In higher dimensions besides black holes with a spherical horizon topology also black objects with different horizon topology exist. In more that 5 dimensions, rapidly rotating black holes develop instabilities, where new branches of pinched black holes appear, which are supposed to be connected with branches of black rings via a merger solution, where the horizon topology changes. New similar families of solutions will be constructed, and followed towards the horizon topology changing transition. At the same time the evolution of the singularity will be monitored. Stability considerations will, however, also be applied to other solutions in general relativity or in generalized models, also including various types of matter fields, and to astrophysically relevant objects.