Semisimple case: Belavin-Drinfeld result does not classify all lie bialgebras on a (finite-dim.) semisimple complex Lie algebra, but only so called quasi-triangular Lie bialgebras (those having a non skew-symmetric r-matrix which satisfies CYBE). If one moves from the quasi-triangular case to the triangular case it can be easily shown that classification of triangular Lie bialgebra structures contains, as a subproblem, that of determining all Frobenius Lie subalgebras, ie. Lie subalgebras such that there exists a linear functional $l$ on them so that the bilinear form $l([X,Y])$ is nondegenerate. Semisimple Lie algebras are not Frobenius but may contain many (non semisimple) Frobenius Lie subalgebras. Unfortunately (and I think Alexander Chervov is referring to this fact) the classification of all Lie subalgebras of a given Lie algebra is a wild problem. See When is a classification problem "wild"?

You can find a neat explanation of all this in Korogodski-Soibelman Algebras of functions on Quantum Groups Part I, Math. Surv and Monographs 36, AMS 1998.

General case: in the general, not even semisimple case, still there are a bunch of interesting results. Lie bialgebra structures on 3-dim Lie algebras have been classified (in fact this result is periodically republished...). I would credit for this Xavier Gomez, Journ. Math. Phys. 41 (2000).

Lie bialgebras were also classified on some specific Lie algebras like Heisenberg-Lie (Christian Ohn in dimension 3 and Andrè Diatta in general, I think - don't know if this last result were published apart from his PhD thesis).

I've seen results also on classifying all Lie bialgebra structures on 4-dim Lie algebras but the list is huge and not very illuminating.

Semisimple case: Belavin-Drinfeld result does not classify all lie bialgebras on a (finite-dim.) semisimple complex Lie algebra, but only so called quasi-triangular Lie bialgebras (those having a non skew-symmetric r-matrix which satisfies CYBE). If one moves from the quasi-triangular case to the triangular case it can be easily shown that classification of triangular Lie bialgebra structures contains, as a subproblem, that of determining all Frobenius Lie subalgebras, ie. Lie subalgebras such that there exists a linear functional $l$ on them so that the bilinear form $l([X,Y])$ is nondegenerate. Semisimple Lie algebras are not Frobenius but may contain many (non semisimple) Frobenius Lie subalgebras. Unfortunately (and I think Alexander Chervov is referring to this fact) the classification of all Lie subalgebras of a given Lie algebra is a wild problem. See When is a classification problem "wild"?

You can find a neat explanation of all this in Korogodski-Soibelman Algebras of functions on Quantum Groups Part I, Math. Surv and Monographs 36, AMS 1998.

General case: in the general, not even semisimple case, still there are a bunch of interesting results. Lie bialgebra structures on 3-dim Lie algebras have been classified (in fact this result is periodically republished...). I would credit for this Xavier Gomez, Journ. Math. Phys. 41 (2000).

Lie bialgebras were also classified on some specific Lie algebras like Heisenberg-Lie (Christian Ohn in dimension 3 and Andrè Diatta in general, I think - don't know if this last result were published apart from his PhD thesis).

I've seen results also on classifying all Lie bialgebra structures on 4-dim Lie algebras but the list is huge and not very illuminating.