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Future galaxy clustering surveys will probe small scales where nonlinearities become important. Since the number of modes accessible on intermediate to small scales is very high, having a precise model at these scales is important especially in the context of discriminating alternative cosmological models from the standard one. In the mildly nonlinear regime, such models typically differ from each other, and galaxy clustering data will become very precise on these scales in the near future. As the observable quantity is the angular power spectrum in redshift space, it is important to study the effects of nonlinear density and redshift space distortion (RSD) in the angular power spectrum. We compute nonlinear contributions to the angular power spectrum using a flat-sky approximation, and compare the results of different perturbative and nonperturbative approaches. We find that the Taruya-Nishimichi-Saito (TNS) perturbative approach is significantly closer to the comoving Lagrangian acceleration approximation than Eulerian or partially resummed-Lagrangian one-loop approximations, effective field theory of large scale structure or a halofit-inspired model. However, none of these prescriptions agree with each other in the nonlinear regime. A surprising and new result of the present analysis is that for narrow redshift bins, Δz≲0.01, the angular power spectrum acquires nonlinear contributions on all scales, right down to ℓ=2, and is hence not a reliable tool at this time. To overcome this problem, we need to model nonlinear RSD terms, for example as TNS does, but for a matter power spectrum that remains reasonably accurate well into the deeply nonlinear regime, such as halofit. [ABSTRACT FROM AUTHOR]