Rapid compression machines (RCMs) are versatile experimental facilities that provide engine-relevant conditions under well-defined boundary conditions. However, they have barely been used to study the laminar flame propagation under such conditions. Therefore, this study examines the applicability of RCMs for the investigation of laminar burning velocities (LBVs). Disturbances induced by the ignition electrodes are studied with hot wire measurements. The findings suggest that in the present investigation the ignition should not occur sooner than ≈76 ms after end of compression (EOC) for a compression phase of around 29 ms as flow field disturbances have subsided to a minimum at this time. Furthermore, the evaluation of flame speed should not last more than 200 ms after EOC as the roll-up vortex then starts to disturb the homogeneous temperature distribution. The experimental procedure is described taking into account the development of cellular unstable flames for hydrogen-containing blends. In addition, an uncertainty quantification has been conducted and uncertainties in the LBV of ±6% have to be expected. Finally, LBVs have been derived with the RCM for neat methane and ammonia flames as well as equimolar blends consisting of these fuels and hydrogen. The investigated equivalence ratio ranged from 0.6 to 1.3 and included pressures at 4 and 9 bar. With a fixed compression ratio, the compressed gas temperature depended on the investigated gas mixture and reached from 618 to 751 K. Different kinetic mechanisms have been compared to the obtained data, and the C3MechV3.5 revealed the best overall predictions with a mean error of −2% and a standard deviation of ±7%. However, model inconsistencies of the LBV results remain with regard to an overprediction of the hydrogen-ammonia blends and a slight underprediction of the hydrogen-methane blends. In conclusion, the RCM seems to be an appropriate facility for the investigation of LBVs.