Smart headphones or hearables use different types of algorithms such as noise cancelation, feedback suppression, and sound pressure equalization to eliminate undesired sound sources or to achieve acoustical transparency. Such signal processing strategies might alter the spectral composition or interaural differences of the original sound, which might be perceived by listeners as monaural or binaural distortions and thus degrade audio quality. To evaluate the perceptual impact of these distortions, subjective quality ratings can be used, but these are time consuming and costly. Auditory-inspired instrumental quality measures can be applied with less effort and may also be helpful in identifying whether the distortions impair the auditory representation of monaural or binaural cues. Therefore, the goals of this study were (a) to assess the applicability of various monaural and binaural audio quality models to distortions typically occurring in hearables and (b) to examine the effect of those distortions on the auditory representation of spectral, temporal, and binaural cues. Results showed that the signal processing algorithms considered in this study mainly impaired (monaural) spectral cues. Consequently, monaural audio quality models that capture spectral distortions achieved the best prediction performance. A recent audio quality model that predicts monaural and binaural aspects of quality was revised based on parts of the current data involving binaural audio quality aspects, leading to improved overall performance indicated by a mean Pearson linear correlation of 0.89 between obtained and predicted ratings.