I've had my share of minor mishaps- the most expensive was a wrought Iron fence section that I crushed taking the very last limb out of a large oak removal. The limb was on the opposite side of the fence, as I had rigged out the other side of the tree. Tried to snap cut the last piece, it hit brush first and catapaulted back onto the fence.
The most frustrating was another oak removal, as I dropped the spar onto the ground the butt end severed an underground cable tv line that was less than an inch deep. Customer came out and asked what we did to his tv. Luckily the cable co didn't charge us to repair it.
I think the biggest thing is, when something happens, no matter how minor, immediately notify and apologize to the homeowner. Let them know you will do whatever it takes to make it right again.
Never, ever, ever, try to cover something up as if it didn't happen or you didn't do it.