Can I bring a flat iron in my carry-on onto the plane??

Yes, according to the TSA website. Always check tsa.gov if you have a question about items in luggage

You may want to also be aware that the TSA at different airports tend to follow different interpretations of the TSA rules. So what goes through the line just fine at SFO may not get through at Denver. Always remember the overriding rule when dealing with the TSA: if you want to make your flight, don't argue, even if you know what the rules allow. You can win by standing on your rights or their policy, but you still will miss your flight.