Nonlinear dynamics of discontinuous oscillators with unilateral constraints and non-random parametric uncertainties are investigated. Nonlinear oscillators considering single- and double-sided constraints are carefully constructed to exhibit rich bifurcations, such as period-doubling and Neimark-Sacker bifurcations. In deterministic amplitude-frequency responses, both hardening and softening effects are induced by non-smooth contact-type nonlinearities. Stabilities of the solutions are determined by the shooting method and the monodromy matrix. To effectively quantify the behaviors of nonlinear oscillators in the presence of parametric uncertainties, a non-intrusive surrogate function aided by arc-length ratio interpolation is constructed. Simulation results demonstrate variabilities of nonlinear responses under different non-random uncertainties. Moreover, an accuracy verification is provided to verify the effectiveness of the non-intrusive uncertainty propagation method. It is found that the surrogate function in combination with the arc-length ratio technique has high accuracy and evolutions of turning points are captured satisfactorily regardless of complex interactions of nonlinearities and uncertainties. The findings and methodologies reported are meaningful to general nonlinear systems having complex motions, paving the road for more in-depth investigations into uncertain nonlinear dynamics.