We consider spatial hysteresis and modulational instability in arrays of nonlinear metallic nanoparticles. We show that such plasmonic systems are characterized by a bistable response, and they can support the propagation of dissipative switching waves (or plasmonic kinks) connecting the states with different polarization. We demonstrate that modulational instability, also inherent in our system, can lead to the formation of regular periodic or quasi-periodic modulations of the polarization. We reveal that arrays of metallic nanoparticles can support nonlinear localized modes of two different types - plasmon-solitons and plasmon-oscillons. They both possess deeply subwavelength size. However, the profile of plasmon-solitons is stationary; whereas plasmon-oscillons has the oscillating profile which can stand at rest or slowly drift along the chain.