Magnetic zigzag edges of graphene are considered as a basis for novel spintronics devices despite the fact that no true long-range magnetic order is possible in one dimension. We study the transverse and longitudinal fluctuations of magnetic moments at zigzag edges of graphene from first principles. We find a high value for the spin wave stiffness D=2100 meV angstrom(2) and a spin-collinear domain wall creation energy E-dw=114 meV accompanied by low magnetic anisotropy. Above the crossover temperature T-x approximate to 10 K, the spin correlation length xi proportional to T-1 limits the long-range magnetic order to similar to 1 nm at 300 K while below T-x, it grows exponentially with decreasing temperature. We discuss possible ways of increasing the range of magnetic order and effects of edge roughness on it.