Total variation regularization is a valuable tool for a wide array of applications ranging from inverse problems to optimal control and machine learning. This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with well-behaved jumpsets. On the downside, their intricate properties significantly complicate every aspect of their analysis, from the derivation of first-order optimality conditions to their discrete approximation and the choice of a suitable solution algorithm. In this talk, we present two accelerated conditional gradient algorithms for this specific problem based on different, constrained surrogate formulations. The resulting methods rely on iterates given by conic combinations of characteristic functions of sets of finite perimeter. In every iteration, the algorithms alternate between proposing new sets to add to the iterate based on the resolution of prescribed curvature problems and finite-dimensional LASSO-like problems to update the associated weights. We discuss their convergence properties, starting from a global, sublinear rate of convergence of the residual. Assuming that the unique minimizer is given by a finite linear combination of characteristic functions as well the strict stability of its level sets, e.g. by coercivity of a suitable shape Hessian, asymptotic linear rates are provided. We briefly describe its practical realization relying on a combination of off-the-shelf graph cut algorithms and warmstarted semismooth Newton methods. The talk is concluded by numerical experiments for PDE-constrained optimization problems, highlighting the practical relevance of the proposed method.