I have often wondered what the official stance on this is, but it turns out there’s a simple page on the TSA website that tells you the answer. Clearly it says that monopods and tripod are ok to travel with, as long as they fall within luggage dimensional limits.
Of course those vary from airline to airline, but roughly speaking I’d say that it gives you room for a 21″ long tripod, as long as when strapped to the side of your bag it doesn’t then push your other bag dimensions over the limit. Seems reasonable.
The CATSA (Canadian version of the TSA) also has a similar page on their website that says the same thing. So those are the rules in N.America at least. Please make sure you check with other countries if you’re leaving the continent! Have you ever flown with a tripod?