We revive an old proposal of Zeh for the preferred basis in the many-worlds interpretation of quantum mechanics. The algorithm for the basis reduces to the eigenvalue problems for density matrices of subsystems forming the whole system under consideration. We generalize this procedure to the case of degenerate eigenvalues of reduced density matrices. A semiclassical calculational method for these eigenvalues is developed and applied to some model problems. The classical properties of elements of the preferred basis are investigated. It is shown that classicality exists only in some part of many-worlds branches. Moreover, it depends crucially on the initial conditions and Hamiltonians and under some circumstances turns out to be a temporary phenomenon. Applications of the preferred-basis proposal to quantum cosmology are discussed. The relation between the preferred-basis approach and quantum-histories approach is discussed.