We determine the phase diagram of LaFe$_{1-x}$Co$_x$AsO 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 LaFe$_{1-x}$Co$_x$AsO are identical to those in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, suggesting that nematicity in LaFeAsO is a result of the fluctuating spin density wave as well.
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