We study how smooth solutions of the Camassa-Holm equation evolve through and beyond the first wave-breaking time. For a broad open class of smooth initial data, the first singularity occurs at a unique point and has the sharp local H\"older regularity \(C^{3/5}\). We show that the first singularity does not simply persist after breaking; instead, it bifurcates immediately into two distinct singular branches forming a cuspon--anticuspon pair. At the breaking time, the solution remains bounded and continuous and loses \(C^1\)-regularity at exactly one point. For every sufficiently small positive time after breaking, this single singularity splits into two cusp-type singularities, each having the sharp local H\"older regularity \(C^{2/3}\). One branch is a cuspon, whose spatial derivative tends to \(+\infty\) and \(-\infty\) from the two sides, while the other is an anticuspon with the opposite orientation. The two singular branches move apart from the original breaking point, and the portion of the profile between them reverses its local monotonicity.
