A singularity is said to be exceptional (in the sense of V. Shokurov), if f
or any log canonical boundary, there is at most one exceptional divisor of
discrepancy -1. This notion is important for the inductive treatment of log
canonical singularities. The exceptional singularities of dimension 2 are
known: they belong to types E-6, E-7, E-8 after Brieskorn. In our previous
paper, it was proved that the quotient singularity defined by Klein's simpl
e group in its 3-dimensional representation is exceptional. In the present
paper, the classification of all the three-dimensional exceptional quotient
singularities is obtained. The main lemma states that the quotient of the
affine 3-space by a finite group is exceptional if an only if the group has
no semiinvariants of degree 3 or less. It is also proved that for any posi
tive epsilon, there are only finitely many epsilon-log terminal exceptional
3-dimensional quotient singularities.