  The Eisenbud--Goto conjecture states that $\operatorname{reg}
X\le\operatorname{deg} X -\operatorname{codim} X+1$ for a nondegenerate
irreducible projective variety $X$ over an algebraically closed field. While
this conjecture is known to be false in general, it has been proven in several
special cases, including when $X$ is a projective toric variety of codimension
$2$. We classify the projective toric varieties of codimension $2$ having
maximal regularity, that is, for which equality holds in the Eisenbud--Goto
bound. We also give combinatorial characterizations of the arithmetically
Cohen--Macaulay toric varieties of maximal regularity in characteristic $0$.
