People here often post questions about flying domestically when they're at various beginning stages of the immigration process. Answers vary from "carry your complete current documentation with you" to "you will have no problems."
Customs & Border Protection agents sometimes hang out at the TSA airport checkpoints. In August 2015, I flew out of a border-area airport in south Texas and took the opportunity to query the CBP agents at the TSA checkpoint.
T-B.: When you hang out at the TSA checkpoints, are you checking that
foreign travelers are in status?
T-B.: For example, what if a K-1 entrant's I-94 is expired?
CBP: We have all kinds of ways of checking their status.
T-B.: Would it help if they have their marriage certificate with them?
T-B.: If they've applied for AOS and have their I-797 or at least a copy of their
cover-letter for their I-485, does it help?
CBP: Absolutely anything that they can provide will help a lot, and will speed things up.
T-B.: Why are you at some airports and not others?
CBP: We're at border airports and others that are ports of entry. We're not at all of them.
T-B.: But it still makes sense for legal immigrants to carry their current documentation?
CBP: Absolutely. It helps them and it helps us. It speeds things up a lot.
T-B.: I'll pass this along. Thank you.
This confirms that it makes sense to carry your most recent documentation with you until you have your green card.
I first had Mrs. T-B. carry her marriage certificate (MC) and passport with valid I-94.
Then, she carried her MC, passport, and copy of her cover-letter for AOS/EAD/AP.
Then, she carried her MC, passport, and her NOA1 (I-797) for AOS.
Then, she carried her MC, passport, and AP.
Then, she carried her first green card (= the law).
Then, she carried that green card and her cover-letter for ROC.
Then, she carried that green card and her NOA1 for ROC.
Then, she carried her 10-year green card until she became a naturalized citizen (= the law).
These documents came in handy at various stages, including at a Texas/Mexico land-crossing POE, a couple of airports, and especially inland CBP checkpoints while driving. At these last, the CBP agents are typically in poor humor and are looking to nail someone. For example, if you're not carrying your green card, they'll detain you for as long as it takes to verify that you're who you say you are.
The above is presented in hopes that it will help others.