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 A2 and a spin-collinear domain wall creation energy E(dw)=114 meV accompanied by low magnetic anisotropy. Above the crossover temperature T(x) approximately 10 K, the spin correlation length xi proportional, variantT(-1) limits the long-range magnetic order to approximately 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.