The similar sublattices of a planar lattice can be classified via its
multiplier ring. The latter is the ring of rational integers in the generic
case, and an order in an imaginary quadratic field otherwise. Several classes
of examples are discussed, with special emphasis on concrete results. In
particular, we derive Dirichlet series generating functions for the number of
distinct similar sublattices of a given index, and relate them to various zeta
functions of orders in imaginary quadratic fields.