In this paper, we present a fully distributed method that allows a group of magnetically connected modular robots to assess the mechanical stability of their structure. Stability is verified based on four mechanical phenomena: vertical and rotational sliding and rotational debonding for vertical and lateral connectors. We present a mechanical model applicable to the Blinky Blocks robot system. We also propose an efficient, fully distributed algorithm that runs simultaneously on all robots, enabling them to check global stability and detect the type and the breakage positions. Our algorithmic solution avoids the global resolution of the system and is effective for both free-loop and complex loop or multi-loop configurations; for the latter, it systematically enumerates all possible spanning trees and traverses each one to assess the stability of the modular structure. The approach was validated both in simulation with VisibleSim and on real Blinky Blocks hardware. Experiments demonstrate accurate and robust detection of all four breakage types in a wide variety of configurations, with low computational overhead and excellent scalability. These results confirm that distributed spanning-tree-based analysis provides an effective alternative to solving global equilibrium equations in modular robotics, enabling reliable structural integrity assessment even in large and highly connected assemblies.