The detection of the theoretically expected dark matter is central to particle physics cosmology. Current fashionable supersymmetric models provide a natural dark matter candidate which is the lightest supersymmetric particle (LSP). Such models conmbined with fairly well understood physics like the quark substructure of the nucleon and the nuclear form factor and/or the spin response function of the nucleus, permit the evaluation of the event rate for LSP-nucleus elastic scattering. The thus obtained event rates are, however, very low or even undetectable. So it is imperative to exploit the modulation effect, i.e. the dependence of the event irate on the earth's annual motion. In this paper we study such a modulation effect both in non directional and directional experiments. We calculate both the differential and the total rates using symmetric as well as asymmetric velocity distributions. We find that in the symmetric case the modulation amplitude is small, less than 0.07. There exist, however, regions of the phase space and experimental conditions such that the effect can become larger. The inclusion of asymmetry, with a realistic enhanced velocity dispersion in the galactocentric direction, yields the bonus of an enhanced modulation effect, with an amplitude which for certain parameters can become as large as 0.46