Airports in the US seem to be very tolerant of carry on bags with fly rods, reels, lines, and flies with modest hook sizes (< 1/0). I've really not had any issues with TSA. Packing gear in rod tubes or carry cases that easily fit into overhead luggage bins helps.
However, once you enter security screening in another country the rules are different. Mexico is very strict about no carry on of fly hooks, fly lines, backing, sharp pliers, even small Phillips head screw drivers. I've heard it is the same traveling in South America.
Also, your carry on bags will be rescreened by security once you enter the airport of a country you are transiting. Dubai, UAE has never given me trouble with my fly fishing gear until this year. I had two back-up fly lines confiscated by security staff there (a mean woman covered in a burka). Deeper in my bag rested my best bonefish and giant trevally reels fully lined up. I distracted her with my best jedi mind trick craft and escaped with both reels intact. Allah is merciful.