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We prove that a singular real-analytic Levi-flat hypersurface in being Segre-degenerate at a point is equivalent to the existence of a so-called support curve, that is, a holomorphic curve that intersects at exactly one point, which in turn is equivalent to the existence of support curves on at least two sides of at . The existence of such two-sided support provides families of analytic discs attached to that covers a neighborhood of . The existence of such discs has two corollaries. First, any function holomorphic on a neighborhood of a Segre-degenerate extends to a fixed neighborhood of . Second, the rational hull of is a neighborhood of , and thus no Levi-flat Segre-degenerate hypersurface in can be rationally convex.