Strongly confined colloidal dispersions under shear can exhibit a variety of dynamical phenomena, including depinning transitions and complex structural changes. Here, we investigate the behavior of such systems under pure oscillatory shearing with shear rate γ[over ̇](t)=γ[over ̇]_{0}cos(ωt), as it is a common scenario in rheological experiments. The colloids' depinning behavior is assessed from a particle level based on trajectories, obtained from overdamped Brownian dynamics simulations. The numerical approach is complemented by an analytic one based on an effective single-particle model in the limits of weak and strong driving. Investigating a broad spectrum of shear rate amplitudes γ[over ̇]_{0} and frequencies ω, we observe complete pinning as well as temporary depinning behavior. We discover that temporary depinning occurs for shear rate amplitudes above a frequency-dependent critical amplitude γ[over ̇]_{0}^{crit}(ω), for which we attain an approximate functional expression. For a range of frequencies, approaching γ[over ̇]_{0}^{crit}(ω) is accompanied by a strongly increasing settling time. Above γ[over ̇]_{0}^{crit}(ω), we further observe a variety of dynamical structures, whose stability exhibits an intriguing (γ[over ̇]_{0},ω) dependence. This might enable new perspectives for potential control schemes.