We consider the solution of large-scale least squares problems where the coefficient matrix comes from the discretization of an ill-posed operator and the right-hand size contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution. We pose the regularization problem as a trust-region subproblem and solve it by means of a recently developed method for the large-scale trust-region subproblem. We present numerical results on test problems, an inverse interpolation problem with real data, and a model seismic inversion problem with real data.