My large suitcase arrived in ZRH with one wheel (of four) missing. The Swissport desk said they couldn't produce a damage report as "missing wheel" does not count as damage under Swiss rules. Apparently it's only damage if the case has a tear (!). They suggested I try filing a customer complaint when I got home, via Swiss' website. There was no Swiss airline desk to complain to, so I hurried off for my train.
I took a few pictures of the case on the train and filed a complaint online after 2 days, with little optimism.
Took 2 weeks for a response but, to my surprise, they told me to take the case to get repaired and send them the bill. If it was not repairable, then I should send them my original receipt for the case, and they would reimburse this this price, less 10% depreciation for each year of age. (I have no receipts for the suitcase and in fact it was bought second hand 3 years' ago, so I don't even know how old it is.)
I took the case to this place on Langestrasse in Zurich (found through a very old post on this forum):
Very friendly place, seems they do repairs and reports for airlines and insurance companies all the time. Dropped the case off one evening and the next day they rang to say it could not be repaired, due to being an unusual brand of case and wheel type. Schirm Fredi produced a report (cost: CHF 25) saying "unrepairable" and confirming the new price of the case in Switzerland as CHF 149. They advised me to send this to the airline.
So I sent all the paperwork off to Swiss along with my bank details, and about 3 days later, they paid CHF 174 into my bank account! After the initial complaint, my emails to Swiss were all answered within 24-48 hours.
Overall it was quite inconvenient travelling into central Zurich twice with the case, and now have to figure out how to dispose of the old case. However, just wanted to share that it is worth complaining online, even if you have no success at the airport.