The spatial correlations of the monomer displacements are studied via molecular-dynamics simulations of a melt of fully flexible, unentangled polymer chains with different length, interacting potential, density, and temperature. Both the scalar and the vector characters of the correlations are considered and their extension quantified in terms of suitable dynamical correlation lengths. Displacements performed at both short, i.e., vibrational, and long times, i.e., comparable to the structural relaxation time, are investigated. On both time scales the spatial correlations are modulated according to the radial distribution function g(r) to an extent which is determined by the character of the correlations, the time scale of the displacements and the structural slowing down. The spatial correlations of the short-time displacements have clear directional character. The modulus correlations of the long-time displacements are more marked, especially for sluggish states. Analogous findings are found by experiments on colloids. By inspecting the dynamical heterogeneities of states with slowed-down dynamics, it is observed that fast monomers exhibit correlations which are stronger and more differing from the bulk than the slow ones. It is shown that states with identical average vibrational monomer displacement exhibit identical spatial correlations of the monomer displacements pertaining to the subsets of the fast and the slow monomers characterizing both the short-time and the long-time dynamical heterogeneities.