  We first investigate concentration and vanishing phenomena concerning Moser
type inequalities in the whole plane which involve complete and reduced Sobolev
norms. In particular we show that the critical Ruf inequality is equivalent to
an improved version of the subcritical Adachi-Tanaka inequality which we prove
to be attained. Then, we consider the limiting space
$\mathcal{D}^{1,2}(\mathbb{R}^2)$, completion of smooth compactly supported
functions with respect to the Dirichlet norm $\|\nabla\cdot\|_2$, and we prove
an optimal Lorentz-Zygmund type inequality with explicit extremals and from
which can be derived classical inequalities in $H^1(\mathbb{R}^2)$ such as the
Adachi-Tanaka inequality and a version of Ruf's inequality.
