  This paper studies the $L^{p}$ boundedness of bilinear Fourier multipliers in
the local $L^{2}$ range. We assume a H\"{o}rmander condition relative to a
singular set that is a finite union of Lipschitz curves. The H\"{o}rmander
condition is sharp with respect to the Sobolev exponent. Our setup generalizes
the non-degenerate bilinear Hilbert transform but avoids issues of uniform
bounds near degeneracy.
