Many problems in scientific computing require minimizing nonsmooth
optimization problems. In many applications, it is common to minimize
the sum of a smooth nonconvex function and a nonsmooth convex function.
For example, imaging  and data science applications require minimizing a
measure of data misfit plus a sparsifying L1- or total-variation
regularizer.  We develop a novel inexact trust-region method to minimize
this nonsmooth regularized problem class. Our method is unique in that
permits and systematically controls the use of inexact objective
function and derivative evaluations while maintaining global convergence
guarantees. Provided one can compute the proximal mapping of the
nonsmooth objective function, our method is a simple modification to the
traditional trust-region algorithm for smooth unconstrained
optimization.  Moreover, when using a quadratic Taylor model for the
trust-region subproblem, our algorithm is an inexact, matrix-free
proximal Newton-type method that permits indefinite Hessians, suitable
for solving large-scale problems. We prove global convergence of our
method in Hilbert space and demonstrate its efficacy on examples from
data science and PDE-constrained optimization.