In your picture it appears that the wood used for the servo mounts is square balsa stock exposing side grain upwards. That means that the screws are separating the grain when tightened - weakening the mount - offering little resistance to hold the screws.

I've been using a layered servo mount - a piece of balsa covered with 1/8" aircraft ply (not lite ply). The ply will maintain the integrity of the "bite" over a long period of time and use. Like others have advised, drill a pilot hole, screw in the servo screw, remove screw and apply thin CA to harden.

Perhaps part of your preflight should include testing the torque on all of your servos. If any are loose, tighten. If they won't tighten - don't fly.