In this paper, two new trust-region algorithms for the numerical solution of systems of nonlinear equalities and inequalities are introduced. The formulation is free of arbitrary parameters and possesses sufficient smoothness to exploit the robustness of the trust-region approach. The proposed algorithms are one-sided least-squares trust-region algorithms. The first algorithm is a single-model algorithm, and the second one is a multi-model algorithm where the Cauchy point computation is a model selection procedure. Global convergence analyses for the two algorithms are presented. Our analysis generalizes to nonlinear systems of equalities and inequalities the well-developed theory for nonlinear least-squares problems. Numerical experiments on the two algorithms are also presented. The performances of the two algorithms are reported. The numerical results validate the effectiveness of our approach.