Personally I don’t carry my GC, because I’m far more likely to lose it than for anyone to ever ask for it. And it’s kind of a nightmare to replace, takes awhile and you can’t travel.
To put it another way, the $100 fine is less expensive to me than the consequences of losing it.
They can also put you in jail for up to 30 days if they're feeling like it. Seems crazy to me but also perfectly legal for them to do.
If I understand the comment above correctly, one can carry an I94 too and those you can just print on the website [1]
I'm guessing there's no size requirements? If so it might be a good idea to just print four on an A4, and have one folded in your wallet, one tucked in your bag-pack, one in the car, etc.
[1]: https://i94.cbp.dhs.gov/search/recent-search