We have had to show our c-cards with every op we've dived with, as far as I can remember, and nitrox cards for nitrox fills except at our local LDS, who issued said card. We've only had to show the log once, in Australia. Most dive ops ask for number of dives and date of last dive, which I usually refer to my log book for unless it was in the last few days.
I have several levels of c-cards and keep them in different places--one in my wallet, the rest in my log book, so if something gets lost I will have others. It is also possible for a dive op to get confirmation of your certification directly from the agency via internet, if necessary. I've never had to do that but I've seen it done. They may charge you for the fax but it's worth it if it saves your trip.