Lasers with high coherence and low frequency and amplitude noise are essential for applications requiring high-precision measurement. Laser frequency stabilization is therefore a critical component of lasers used in these applications. In this work, we investigate the frequency stability characteristics of a laser system utilizing Pound-Drever-Hall (PDH) frequency locking and two different frequency discrimination systems, one based on a free-space Fabry-Pérot (F-P) cavity and another based on fiber Bragg gratings (FBGs). The result shows that F-P-based systems offer better performance than FBG-based systems, with each system showing frequency drifts of 83 MHz and 197 MHz respectively over a period of 30 min. The advantages and limitations of both approaches are discussed.