We determine the phase diagram of LaFe1−xCoxAsO single crystals by using nuclear magnetic resonance (NMR). Up to a nominal doping of x = 0.03, it follows the phase diagram for F-doped polycrystals. Above x = 0.03, the F-doped samples become superconducting, whereas for Co-doping the structural and magnetic transitions can be observed up to x = 0.042, and superconductivity occurs only for higher doping levels and with reduced transition temperatures. For dopings up to x = 0.056, we find evidence for short-range magnetic order. By means of relaxation-rate measurements, we map out the magnetic fluctuations that reveal the interplay of nematicity and magnetism. Above the nematic ordering, the spin fluctuations in LaFe1−xCoxAsO are identical to those in Ba(Fe1−xCox)2As2, suggesting that nematicity in LaFeAsO is a result of the fluctuating spin density wave as well.