Nanoscale pores exhibit transport properties that are not seen in micrometre-scale pores, such as increased ionic concentrations inside the pore relative to the bulk solution, ionic selectivity and ionic rectification. These nanoscale effects are all caused by the presence of permanent surface charges on the walls of the pore. Here we report a new phenomenon in which the addition of small amounts of divalent cations to a buffered monovalent ionic solution results in an oscillating ionic current through a conical nanopore. This behaviour is caused by the transient formation and redissolution of nanoprecipitates, which temporarily block the ionic current through the pore. The frequency and character of ionic current instabilities are regulated by the potential across the membrane and the chemistry of the precipitate. We discuss how oscillating nanopores could be used as model systems for studying nonlinear electrochemical processes and the early stages of crystallization in sub-femtolitre volumes. Such nanopore systems might also form the basis for a stochastic sensor.