My question too...like I said, I have to carry my I-551 (Green Card) with me at all times, and to replace it costs $450...
I keep electronic scans of my certificates, mainly so that if I lose the originals, I have all the details (date of issue, certificate number, etc.) from the original to speed the process of applying for appropriate replacements. That said, if I ever found myself caught short and ramped when I accidentally left my wallet in the car at another airport, I could pull up the e-Copy in less than a minute and beg for mercy!
In the 15 years I had a Green Card, the only time I ever carried it was coming into the US. As for e copies I do them as well, actually I have them on my website, if they or I have a smart phone or tablet I can show them a facimile they can verify, I also have all my documents individually as jpgs on my phone and ipad through my email account. I can access this stuff six ways from Sunday electronically and I've been doing it since the first Siemens Pocket PC Phone Edition came out. It's just not a big deal.
The reality is it went like this, I was flying to Oxnard with the owners wife in the right seat and buddy in the back, no luggage but we do have a few hundred pounds worth of anchor and chain. I just grabbed my case but didn't look in it since my head sets hang and my plane stayed ready to fly. Took off out of LGB on 25L transfered to LAX, noticed an oil leak on port, caged it, "LA 04Y caged an engine at 1300' turning back Long Beach" "Roger, good luck","Long Beach 04Y returning 25L down one engine""Roger, would you prefer 07R?""No thanks, I'm holding 1300 10 over blue with power to spare, I'll take the left turns to parking.""Roger" entering downwind..."04Y, would you like me to roll the equipment?""Sure, I might mess this up yet." about 200' on final..."how's that gear indicator looking?""Green locked and down, thanks for thinking of it." Rolled it onto our ramp shy of my parking spot and met with CFR, opened the cowl, identified the problem and that there was no fire hazard and released them. At this time LGBPD working as airport security rolled up, asked me for my documents, I pulled out my captains license, said "oops, wrong case", the kid got in a huff, the sergeant said, "come on, lets go, that isn't our issue". I walked in the shop, called the tower as they requested, gave them my report which took 25 seconds, went and replaced the cracked fitting while Dave drove back to the boat and swapped my cases. We got into Oxnard 2 hrs late, no big deal.
The funniest thing was the owners wife was just her normal relaxed bubbly self the whole time. Dave on the other hand was terrified. When we were back up and flying Dave asked her, "Weren't you scared at all?" her reply? "Shoot no Henning sails us into the dock with no engine all the time she's way bigger than this plane."