A continuously stirred decanting reactor (CSDR) is a well-mixed vessel fed with two immiscible liquid phases, while its effluent consists of a single phase containing the reaction products. We consider a CSDR in which a desired highly exothermic reaction occurs in one phase, while and undesirable exothermic side reaction occurs in the second extracting phase. The holdup of the two phases in the CSDR is sensitive to its temperature. Thus, the reactor has to be kept in a narrow temperature range to avoid breakthrough, that is, with effluent containing both phases, and to maintain a high selectivity of the desired product. Pilot-plant data for a proprietary selective-oxidation process verified the predictions of the steady-state model.A high rate of heat generation may lead to various exotic and undesired periodic and aperiodic oscillations, which may shift the reactor out of the region of safe operation and have deleterious impact on the decantation efficiency. In some cases, the oscillations lead to a breakthrough during part of the period. To avoid these oscillatory states, one may have to maintain a sufficiently small difference between the reactor and coolant temperatures. We show how loci of local and global singular points of codimension 1 and 2 can be used to construct maps of parameters regions with qualitatively different steady-state and dynamic bifurcation diagrams. These maps clearly describe the desirable regions of operation and point out all the potential stability and operability issues in other regions.