Processing circuitry 4 has a plurality of exception states EL0-EL3 for handling exception events, the exception states including a base level exception state EL0 and at least one further level exception state EL1-EL3. Each exception state has a corresponding stack pointer indicating the location within the memory of a corresponding stack data store 35. When the processing circuitry is in the base level exception state EL0, stack pointer selection circuitry 40 selects the base level stack pointer as a current stack pointer indicating a current stack data store for use by the processing circuitry 4. When the processing circuitry 4 is a further exception state, the stack pointer selection circuitry 40 selects either the base level stack pointer or the further level stack pointer corresponding to the current further level exception state as a current stack pointer.