Proteins, nucleic acids and ions secreted from single cells are the key signalling factors that determine the interaction of cells with their environment and the neighbouring cells. It is possible to study individual ion channels by pipette clamping, but it is difficult to dynamically monitor the activity of ion channels and transporters across the cellular membrane. Here we show that a solid-state nanopore integrated in an atomic force microscope can be used for the stochastic sensing of secreted molecules and the activity of ion channels in arbitrary locations both inside and outside a cell. The translocation of biomolecules and ions through the nanopore is observed in real time in live cells. The versatile nature of this approach allows us to detect specific biomolecules under controlled mechanical confinement and to monitor the ion-channel activities of single cells. Moreover, the nanopore microscope was used to image the surface of the nuclear membrane via high-resolution scanning ion conductance measurements.