  It is known that the optimal Noether inequality $\mathrm{vol}(X) \ge
\frac{4}{3}p_g(X) - \frac{10}{3}$ holds for every $3$-fold $X$ of general type
with $p_g(X) \ge 11$. In this paper, we give a complete classification of
$3$-folds $X$ of general type with $p_g(X) \ge 11$ satisfying the above
equality by giving the explicit structure of a relative canonical model of $X$.
This model coincides with the canonical model of $X$ when $p_g(X) \ge 23$. We
also establish the second and third optimal Noether inequalities for $3$-folds
$X$ of general type with $p_g(X) \ge 11$. These results answer two open
questions raised by J. Chen, M. Chen and C. Jiang, and in dimension three an
open question raised by J. Chen and C. Lai. A novel phenomenon shows that there
is a one-to-one correspondence between the three Noether inequalities and three
possible residues of $p_g(X)$ modulo $3$.
