The detection of single nuclear spins would be useful for fields ranging from basic science to quantum information technology. However, although sensing based on diamond defects and other methods have shown high sensitivity, they have not been capable of detecting single nuclear spins, and defect-based techniques further require strong defect-spin coupling. Here, we present the detection and identification of single and remote C-13 nuclear spins embedded in nuclear spin baths surrounding a single electron spin of a nitrogen-vacancy centre in diamond. We are able to amplify and detect the weak magnetic field noise (∼10 nT) from a single nuclear spin located ∼3 nm from the centre using dynamical decoupling control, and achieve a detectable hyperfine coupling strength as weak as ∼300 Hz. We also confirm the quantum nature of the coupling, and measure the spin-defect distance and the vector components of the nuclear field. The technique marks a step towards imaging, detecting and controlling nuclear spins in single molecules.