Piezoresponse force microscopy (PFM) has emerged as a powerful tool to characterize piezoelectric, ferroelectric, and multiferroic materials on the nanometer level. Much of the driving force for the broad adoption of PFM has been the intense research into piezoelectric properties of thin films, nanoparticles, and nanowires of materials as dissimilar as perovskites, nitrides, and polymers. Recent recognition of limitations of single-frequency PFM, notably topography-related cross-talk, has led to development of novel solutions such band-excitation (BE) methods. In parallel, the need for quantitative probing of polarization dynamics has led to emergence of complex time- and voltage spectroscopies, often based on acquisition and analysis of multidimensional datasets. In this perspective, we discuss the recent developments in multidimensional PFM, and offer several examples of spectroscopic techniques that provide new insight into polarization dynamics in ferroelectrics and multiferroics. We further discuss potential extension of PFM for probing ionic phenomena in energy generation and storage materials and devices.