tl;dr: You can fly within the US and also you are good to travel to Canada and Mexico, for 30 days or less.
Your legal status depends on the I-94 date stamp and not the visa. You can retrieve your I-94 at the Department of Homeland Services. Once your I-94 has expired, if you are still in the US, you are overstaying. Nothing I am saying should be considered endorsing that.
If your I-94 has not expired, then you are legally staying within the United States. This includes flying. If you are overstaying, anecdotal evidence suggests that the TSA and airline agents will only check the validity of your passport and not care about visa stamps. They do not even have any legal basis to require you to show the latest visa stamp, as they are not law enforcement. In an ordinary case you won't meet Customs and Border Protection officials. Some airports do not even have any. Obviously, if your behavior suggests a problem exists, then law enforcement can get involved regardless of whether it's an airport or not, but more easier at an airport and then you can get in trouble.
You can check the official travel site for Re-entering the United States with a Valid I-94 Form and Expired Visa:
Non-immigrants who departed the United States for brief travel to Canada, Mexico, or an adjacent island (for F and J non-immigrants) for thirty days or less
This CBP document gives you more details, but it's the same basic provision. It's also much clearer about what the (for F and J non-immigrants) refers to: only the adjacent islands.