In this work radiation reaction (RR), classically described as the backreaction of radiation emitted by an electron on its own dynamics, is investigated in a quantum framework. Theoretical studies in this field are of considerable interest, since unambiguous signatures of RR have not been reported yet and only a thorough theoretical understanding will pave the road towards the desired experimental validation. Here, we investigate signatures of RR in the experimentally highly interesting ultrarelativistic quantum regime and demonstrate possible detection schemes for novel signatures of previously untested physics. In detail, we study the influence of RR effects on the dynamics of electron beams interacting with intense laser pulses, which is found to be significantly altered by quantum effects. Next to this, we demonstrate how the inclusion of higher order photon emissions, in addition to the usually studied single-photon emission, reproduces quantum patterns in the angular distribution of the emitted radiation, which are strongly reminiscent of patterns attributed to classical RR effects. The numerical results presented in this work indicate the accessibility of the nonlinear quantum regime with available electron accelerator and laser technology, opening the possibility of finding answers to long-standing questions of electrodynamics.