  We study the dynamical structure factors of the spin-1/2 XXX chain at finite
magnetic field h, focusing in particular on the singularities at excitation
energies in the vicinity of the lower thresholds of the leading-order dominant
excitations. We derive the exact momentum and spin-density dependences of the
exponents controlling the singularities for both the longitudinal and
transversal dynamical structure factors for the whole momentum range, in the
thermodynamic limit. In that limit we argue that the higher-order excitations
change neither the exponents nor the sharpness of the singularities. We discuss
the relation to neutron scattering and suggest new experiments on spin-chain
compounds using a carefully oriented crystal.
