Nitrogen vacancy (NV) centres in diamond are promising elemental blocks for quantum optics, spin-based quantum information processing and high-resolution sensing. However, fully exploiting the capabilities of these NV centres requires suitable strategies to accurately manipulate them. Here, we use optical tweezers as a tool to achieve deterministic trapping and three-dimensional spatial manipulation of individual nanodiamonds hosting a single NV spin. Remarkably, we find that the NV axis is nearly fixed inside the trap and can be controlled in situ by adjusting the polarization of the trapping light. By combining this unique spatial and angular control with coherent manipulation of the NV spin and fluorescence lifetime measurements near an integrated photonic system, we demonstrate individual optically trapped NV centres as a novel route for both three-dimensional vectorial magnetometry and sensing of the local density of optical states.