The nature of 0⁺ excitations, especially in transitional and deformed nuclei, has attracted new attention. Following a recent experiment studying ¹⁵⁸Gd, we investigated a large group of nuclei in the rare-earth region with the (p,t) pickup reaction using the Q3D magnetic spectrograph at the University of Munich MP tandem accelerator laboratory. Outgoing tritons were recorded at various lab angles, and their angular distributions are compared to those calculated using the distorted-wave Born approximation. Using the unique shape of the L=0 angular distribution, more than double the number of 0⁺ states than were previously known are identified. The distribution of 0⁺ energies and cross sections is discussed in terms of collective and noncollective degrees of freedom, and the density of low-lying 0+ states is discussed as a corroboration of a characteristic feature of phase transition regions. The degree of level mixing, as extracted from Brody distribution fits to the energy spacings of adjacent 0+ levels, is also explored.