The dynamics of cool flames are often described by maps of the boundaries between qualitatively different types of behavior. A scheme is presented for the identification of these boundaries (bifurcations) in the dynamic behavior. This information yields important insight into the behavior of the system and is helpful in mathematical model development and selection. Even more discriminating information is obtained by finding the nature of singular points existing at the intersection of several bifurcation sets. A systematic identification requires a classification of all the possible singular points. Such a classification is presented for a certain class of frequently occurring systems having widely separated time scales. The methodology is illustrated by the analysis of observations in propane and acetadehyde oxidation.