This paper presents a detailed assessment of the ability of the 240 Skyrme interaction parameter sets in the literature to satisfy a series of criteria derived from macroscopic properties of nuclear matter in the vicinity of nuclear saturation density at zero temperature and their density dependence, derived by the liquid drop model, experiments with giant resonances and heavy-ion collisions. The objective is to identify those parameterizations which best satisfy the current understanding of the physics of nuclear matter over a wide range of applications. Out of the 240 models, only 16 are shown to satisfy all these constraints. Additional, more microscopic, constraints on density dependence of the neutron and proton effective mass beta-equilibrium matter, Landau parameters of symmetric and pure neutron nuclear matter, and observational data on high- and low-mass cold neutron stars further reduce this number to 5, a very small group of recommended Skyrme parameterizations to be used in future applications of the Skyrme interaction of nuclear matter related observables. Full information on partial fulfillment of individual constraints by all Skyrme models considered is given. The results are discussed in terms of the physical interpretation of the Skyrme interaction and the validity of its use in mean-field models. Future work on application of the Skyrme forces, selected on the basis of variables of nuclear matter, in Hartree-Fock calculation of properties of finite nuclei, is outlined.

86 pages, 14 figures