In order to improve the stability of label propagation algorithm and reduce the randomness which causes difference in the results of community discovery, labels initialization, random nodes queues setting and labels random selection are improved respectively, and a stable label propagation method for community discovery is proposed. This method first initializes labels by searching for non-overlapping triangles in the networks, and then forms nodes queues based on labels entropy and random sorted nodes in the sub queues. At last, this method chooses labels for each node by the distribution of adjacent nodes labels. Experimental results shows that, stability indexes and quality indexes of our method are higher than other methods' on three social networks—Zachary's Karate club, dolphin social network and American College football. Community discovery based on stable label propagation method not only maintains the advantages of label propagation algorithm, but also improves the quality and stability of community discovery results.