We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the $c=1$ Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge $c$ is in a certain subset of $(1,\infty)$, including $[2,\infty)$, and $h \geq (c-1)/24$, the irreducible representation with lowest weight $h$ of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge $c$ is in the above set and satisfies $c\leq 25$ then the corresponding Virasoro net has no proper local extensions of compact type.