Quasiperiodicity in models of the peroxidase-oxidase reaction has previously been reported in "abstract" or phenomenological models which sacrifice chemical realism for tractability. In the present paper, we discuss how such behavior can arise in a detailed model (BFSO) of the reaction which has previously been shown to be consistent with experimental findings. We distinguish two types of quasiperiodic behavior. Regions of what we here refer to as "primary" quasiperiodicity are delimited by supercritical secondary Hopf bifurcations at one end of the relevant range of parameter values and by heteroclinic transitions at the other. Regions of so-called "secondary quasiperiodicity" are delimited by supercritical Hopf bifurcations at both ends of the parameter range. The existence of a quasiperiodic route to chaos in a modified version of BFSO is also described. This paper emphasizes the experimental circumstances under which quasiperiodic dynamics may be detected in the lab and offers specific prescriptions for its observation.