  A holomorphic 1-form on a compact Riemann surface S naturally defines a flat
metric on S with cone-type singularities. We present the following surprising
phenomenon: having found a geodesic segment (saddle connection) joining a pair
of conical points one can find with a nonzero probability another saddle
connection on S having the same direction and the same length as the initial
one. The similar phenomenon is valid for the families of parallel closed
geodesics.
  We give a complete description of all possible configurations of parallel
saddle connections (and of families of parallel closed geodesics) which might
be found on a generic flat surface S. We count the number of saddle connections
of length less than L on a generic flat surface S; we also count the number of
admissible configurations of pairs (triples,...) of saddle connections; we
count the analogous numbers of configurations of families of closed geodesics.
By the previous result of A.Eskin and H.Masur these numbers have quadratic
asymptotics with respect to L. Here we explicitly compute the constant in this
quqadratic asymptotics for a configuration of every type. The constant is found
from a Siegel--Veech formula.
  To perform this computation we elaborate the detailed description of the
principal part of the boundary of the moduli space of holomorphic 1-forms and
we find the numerical value of the normalized volume of the tubular
neighborhood of the boundary. We use this for evaluation of integrals over the
moduli space.
