  We describe a class $\mathcal{C}$ of punctured torus bundles such that, for
each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are
virtually Haken.
  We show that $\mathcal{C}$ contains infinitely many commensurability classes,
and we give evidence that $\mathcal{C}$ includes representatives of ``most''
commensurability classes of punctured torus bundles.
  In particular, we define an integer-valued complexity function on monodromies
$f$ (essentially the length of the LR-factorization of $f_*$ in
$PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$
has complexity at most 5, then
  $M$ is finitely covered by an element of $\mathcal{C}$.
  If the monodromy has complexity at most 12, then, with at most 36 exceptions,
  $M$ is finitely covered by an element of $\mathcal{C}$.
  We also give a method for computing ``algebraic boundary slopes'' in certain
finite covers of punctured torus bundles.
