Many finite difference methods that involve spatial adaptation employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This constructive strategy has been proven to be extremely effective and easy-to-use in multiphysical computations. However, selections of core monitoring functions are often challenging and crucial to the computational success. This note concerns several different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first a few monitoring designs to be discussed are within the so-called direct regime, the rest belong to a newer category of the indirect type, which requires the priori knowledge of certain important solution features or characteristics. Some simulated examples will be presented to illustrate our study and conclusions. This note is based on recent collaborative work with M. A. Beauregard and J. L. Padgett.