  A lattice in Euclidean $d$-space is called well-rounded if it contains $d$
linearly independent vectors of minimal length. This class of lattices is
important for various questions, including sphere packing or homology
computations. The task of enumerating well-rounded sublattices of a given
lattice is of interest already in dimension 2, and has recently been treated by
several authors. In this paper, we analyse the question more closely in the
spirit of earlier work on similar sublattices and coincidence site sublattices.
Combining explicit geometric considerations with known techniques from the
theory of Dirichlet series, we arrive, after a considerable amount of
computation, at asymptotic results on the number of well-rounded sublattices up
to a given index in any planar lattice. For the two most symmetric lattices,
the square and the hexagonal lattice, we present detailed results.
