A state vector (vector-valued distribution) of an unstable particle can be a 3-momentum eigenvector in at most one inertial frame as a consequence of the necessary width of the energy spectrum. We investigate this frame dependence of momentum eigenvectors of unstable particles and demonstrate that:(i) It is compatible with the principle of relativity.(ii) It leads to a distinction between two ways of defining the spin of the particle which are equivalent for stable particles.(iii) One definition, called kinematical spi~, yields a precise value, and is deter~ined by considerations of detailed balancing and other means of countmg the degeneracy of the momentum eigenvectors.(iv) The second definition, called dynamical spin, need not yield a sharp value, is determined by the dynamics of the decay process, and is measured by observing angular distributions of decay products.It is shown that an unstable particle with vanishing kinematical spin (pion) may have a small admixture of nonisotropic angular distribution of unpolarized decay ~roducts in the rest frame of. the decay pr,oducts: T~e order of magnitude of the effect is estimated and the couplmg between the mass and spm spectrum IS studied m the case of local interactions without derivative coupling.