We consider the energy and the polar angle distributions of J/psi's produced via the color-singlet and the color-octet (S-3(1)(8)) mechanisms in Z(0) --> J/psi + X at LEP. Since both distributions of the J/psi produced via color-octet mechanism are significantly different from those via color-singlet mechanism, these observables can be used as tests of color-octet production mechanism for heavy quarkonia. We also discuss Z(0) --> Y + X and W --> J/psi (or Y) + X in brief.