  We give criteria for the membership of Hankel operators on the Hardy space on
the disc in the Dixmier class, and establish estimates for their Dixmier trace.
In contrast to the situation in the Bergman space setting, it turns out that
there exist Dixmier-class Hankel operators which are not measurable (i.e. their
Dixmier trace depends on the choice of the underlying Banach limit), as well as
Dixmier-class Hankel operators which do not belong to the $(1,\infty)$
Schatten-Lorentz ideal. A related question concerning logarithmic interpolation
of Besov spaces is also discussed.
