  Curve singularities are classical objects of study in algebraic geometry. The
key player in their combinatorial structure is the {\it value semigroup}, or
its compactification, the {\it value semiring}. One natural problem is to
explicitly determine the value semirings of distinguished infinite classes of
singularities, with a view to understanding their asymptotic properties. In
this paper, we establish a matroidal framework for resolving this problem for
singularities determined by arrangements of toric branches; and we obtain
precise quantitative results in the case of line arrangements. Our results have
implications for the topology of Severi varieties of unisingular rational
curves in projective space.
