Motivated by the Chen–Sun description of analytic tangent cones of Hermitian–Yang–Mills connections, we develop a valuation-theoretic framework for algebraic tangent cones of torsion-free sheaves. Replacing the blow-up valuation, which corresponds to the local behaviour of smooth Kähler metrics, by finitely generated valuations, we construct degenerations via Rees algebras and introduce a slope stability theory for the associated graded modules. We define an instability functional and prove the existence of an optimal degeneration for quasi-regular valuations. Moreover, we show that its Harder–Narasimhan graded object is uniquely determined up to grading twists.