I should add one reference, for the case of real numbers, namely the classification 
of all *solvable real six-dimensional Lie algebras* by Turkowsky:

P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.

As mentioned befiore, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to **arbitrary fields**, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.