A speckle-displacement-based wavemeter is combined with a spatial-fundamental-mode-pass filter to eliminate the influence of multimode operation on the directionality of the resulting output from a distributed Bragg reflector (DBR) tapered laser. The proposed setup is characterized theoretically and experimentally, and detections of mode hops and side-mode suppression ratios (SMSRs) in the optical output are demonstrated. The laser illuminates a rough surface at an oblique angle, and a camera observes the corresponding speckle pattern from an almost identical back-scattering direction. As the wavelength of the laser shifts, the speckle pattern responds with a corresponding displacement, which is approximately linear with respect to the shift within the detection area. The wavemeter tracks continuously the shifts of the speckles pattern by tracking the peak of the covariance function of sequentially acquired images. In this way, the speckle-displacement-based wavemeter achieves a spectral resolution of 10.4 MHz. Mode hops in the laser do not cause any impeding decorrelation of the speckle patterns. Interestingly, the actual SMSR is related to the peak height and width of the absolute covariance function. A wavemeter, which is capable of measuring wavelengths, mode hops, and SMSRs, is highly useful for spectroscopy, quantum optics, nonlinear frequency conversion, and other applications requiring stable single-frequency laser light, especially when using diode lasers.