We analyze the scalar field sector of the Kazakov-Migdal model of induced QCD. We present a detailed description of the simplest one-dimensional (d=1) model which supports the hypothesis of wide applicability of the mean-field approximation for the scalar fields and the existence of critical behavior in the model when the scalar action is Gaussian. Despite the occurrence of various nontrivial types of critical behavior in the d=1 model as N→∞, only the conventional large N limit is relevant for its continuum limit. We also give a mean-field analysis of the N=2 model in any d and show that a saddle point always exists in the region m2 > m crit2 (= d). In d=1 it exhibits critical behavior as m2 -> m crit2. However when d>1 there is no critical behavior unless non-Gaussian terms are added to the scalar field action. We argue that similar behavior should occur for any finite N thus providing a simple explanation of a recent result of D. Gross. We show that critical behavior at d>1 and m2 > m crit2 can be obtained by adding a logarithmic term to the scalar potential. This is equivalent to a local modification of the integration measure in the original Kazakov—Migdal model. Experience from previous studies of the Generalized Kontsevich Model implies that, unlike the inclusion of higher powers in the potential, this minor modification should not substantially alter the behavior of the Gaussian model.