While domains of steady-state multiplicity of a single pellet have been mapped for many reactions, systematic studies of the bifurcation set of the whole reactor are missing. Local multiplicity implies communication by propagation of thermal fronts in the bed. This research examines the multiplicity features of a highly exothermic and self-inhibited reaction (ethylene oxidation on supported platinum) conducted in adiabatic and nonadiabatic fixed beds, and compares them to the features of a single pellet freely suspended in a reaction stream or embedded in a few layers of the catalyst. The bistability domain of the latter shows a better resemblance to the multiplicity domain of the adiabatic reactor. Multiplicity of up to three solutions is observed in the fixed-bed reactor while the single pellet exhibits bistability. A steep stationary temperature front exists near the reactor inlet but stationary fronts separating different stable states have not been detected. A narrow domain of bistability is mapped for the nonadiabatic reactor.