We consider the efficient numerical minimization of Tikhonov functionals with nonlinear operators and non-smooth and non-convex penalty terms, which appear e.g. in variational regularization. For this, we consider a new class of SCD semismooth* Newton methods, which are based on a novel concept of graphical derivatives, and exhibit locally superlinear convergence. We present a detailed description of these methods, and provide explicit algorithms in the case of sparsity ($\ell_p$, $0\leq p < \infty$) and TV penalty terms. The numerical performance of these methods is then illustrated on a number of tomographic imaging problems.