Here's my best story about liability:
I visited a client to evaluate a tree. Silver maple, with weak fork, some dead branches, obvious broken spar decaying, with an 8" limb attached extending 20' over his back porch: a curved glass greenhouse ! Naturally, I suggested trimming off the overhang attached to the spar, so that it never could fall on the greenhouse.
Ok, now I am doing the job. Nothing overhead to tie off to, I am hanging out on this branch, and have everything rigged and ready to cut. Prepared about 2' of slack in the bull rope, and gave the 8" limb a nice undercut. Slack still hanging, as I intended, and JUST AS the cut begins to drop, closing the undercut... the groundman yanks out the slack, it hits my chainsaw (I never have figured out how that happened), and clips the line clean off ! No more rope in the tree to tie to, either, as he pulled the rope out of the tree.
Panic stricken, I am seeing thousands of dollars in damage coming my way. Magically, the brisk wind died down to a calm, just as the rope fell. I screamed for another rope to be sent up on my climbing line [I would have used my climbing line first, but the limb was too heavy for me to tend it.]
I got a new line, had to reach WAY out to grab the tag end of the rope still secured to the limb, and quickly tied a square knot. At the very moment I yanked the last tie on the square knot, the hinge broke, and I grabbed the loose line, adding a small amount of friction to pull the falling branch away from the gutters of the house. My groundmen finally managed to grab the rest of the load, and the whole thing swung gracefully past the glass roof. It almost looked like we knew what we were doing all the time.
Smoked my palm good on the rope, but no liability claims.
Apart from an occasional top rail or fence section, That's about it for me.
Oops ! I forgot about that 4' oak log that broke the rope...another day for that story.