A few examples are available as rendered IPython notebooks.
How to take planar slices of a mesh, like you would for 3D printing, using trimesh. Cross sections
Find a path from one mesh vertex to another, travelling along edges of the mesh. Edge graph traversal
Do simple ray- mesh queries, including finding the indexes of triangles hit by rays, the locations of points hit on the mesh surface, etc. Ray queries have the same API with two available backends, one implemented with just Numpy calls, and one that requires pyembree but is 50x faster (a team at Intel wrote it). Ray-mesh queries