We report observations of the spatial and temporal variations in flux penetration into doubly connected bulk superconductors (hollow cylinder, ring) and discs (singly connected) from thermomagnetic avalanches. Dramatic changes were established in the stability of the critical state of the disc when it was transformed into a ring: the region of instability in the magnetic field in the ring doubled, and the number of thermomagnetic avalanches increased from 9 to 60. The duration of the flux jump, the amount of flux entering the sample, and its dependence on the magnetic field changed significantly, and the weak linear field dependence in the disc was inversely proportional to the magnetic field in the ring. A complicated fine structure of the magnetic flux penetrating the ring hole was detected. It was experimentally determined that the speed of the avalanche flux front throughout the sample increased sharply when magnetic flux appeared in the hole. This study also presents other aspects of the dynamic response of a superconducting ring that are discussed in detail.