  We consider flat surfaces and the points of their metric completions,
particularly the singularities to which the flat structure of the surface does
not extend. The local behavior near a singular point x can be partially
described by a topological space L(x) which captures all the ways that x can be
"approached linearly". The homeomorphism type of L(x) is an affine invariant.
When x is not a cone point or an infinite-angle singularity, we say it is wild;
in this case it is necessary to add further metric data to L(x) to get a
quantitative description of the surface near x.
