The Bingham model for viscoplastic materials involves the minimization of a non-differentiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: The total variation minimization problem seeks the unique minimizer $u ∈$ BV$(Ω)$ of bounded variation of the energy $\frac{1}{2} \|u − f\|^2_{L^2(Ω)} + |u|_{BV(Ω)}$ for data $f ∈ L^2(Ω)$ in a bounded Lipschitz domain $Ω ⊂ \mathbb{R}^n$. Our main result proves for a convex domain $Ω$ that $f ∈ H^1(Ω)$ implies $u ∈ H^1(Ω)$. A modification for homogeneous Dirichlet conditions involves an additional trace term $u \in L^1(∂Ω)$ and then $f ∈ H^1_0(Ω)$ implies $u ∈ H^1_0(Ω)$. In the case of the vector Bingham model without viscosity, the boundary conditions are difficult to handle, but we prove the local $H^1_{loc}(Ω)^n$ regularity of the solution for a right-hand side $f ∈ H^1_{loc}(Ω)^n$. The proofs rely on several generalizations of a lemma due to H. Brézis and on the approximation with small viscosity. As a consequence, we obtain Euler-Lagrange characterizations of the solution. Homogeneous Dirichlet conditions on the viscous problem lead in the vanishing viscosity limit to relaxed boundary conditions of frictional type.