This paper presents a system aiming at joint dereverberation and noise reduction by applying a combination of a beamformer with a single-channel spectral enhancement scheme. First, a minimum variance distortionless response beamformer with an online estimated noise coherence matrix is used to suppress noise and reverberation. The output of this beamformer is then processed by a single-channel spectral enhancement scheme, based on statistical room acoustics, minimum statistics, and temporal cepstrum smoothing, to suppress residual noise and reverberation. The evaluation is conducted using the REVERB challenge corpus, designed to evaluate speech enhancement algorithms in the presence of both reverberation and noise. The proposed system is evaluated using instrumental speech quality measures, the performance of an automatic speech recognition system, and a subjective evaluation of the speech quality based on a MUSHRA test. The performance achieved by beamforming, single-channel spectral enhancement, and their combination are compared, and experimental results show that the proposed system is effective in suppressing both reverberation and noise while improving the speech quality. The achieved improvements are particularly significant in conditions with high reverberation times.