These are my experiences:
(7 international flights,

using BA each time)
When Philip was in a rear-facing car seat we put the carseat in with the luggage and didn't use it on the flight. Instead we used the "bassinette" seats the airline provided, which looked a bit like a bouncy seat - it sits in a fold out table in the Cargo area. He was allowed to use it during the flight, but on take off and landing he needed to sit in our laps using the Figure 8 shaped seatbelt that went around we and him. When the "fasten safety belt" signs came on, he needed to be in our lap in those belts as well.
When Philip was in a forward facing seat, we did two things:
1. Brought the seat with us on the plane. We were told by the staff that it needed to be an "approved" 5 point harness in order for him to be allowed to use it during take-off and landing. The take-off and landing bits are where they are sticklers. Ours was a 5 point harness, but the airline attendant who was helping us wasn't *positive* it was "approved" ( there seems to be a lot of confusion on THEIR part, and no one seems to agree! ) so in order to play it safe we were told we had to have him on our lap during take-off and landing (using the special seat belt again) but could use the car seat during the flight. We actually chose to just stow it away (within the plane) at that point because as it turned out he enjoyed using that empty seat as a play space. So whatever.
2. We stored the car seat in with our luggage and collected it after the flight. We did this after doing the above scenario, just because we know what Philip is like and he doesn't like his car seat on the best of days, so is much happier (and less screamy) if he is on our lap. We also knew that if it got bad enough and we could tell he actually DID want to be secured in a seat, we could ask the attendant for their "bassinette" seat. (I don't know why they call it a bassinette seat, it is nothing like a bassinette...

)
This time around (we are flying next weekend, but within the US) I am leaving his car seat in our car, flying him without it, and renting a car on the other end with a car seat already in it. If you are renting a car in the US, consider that as an option as well. You have to call the rental place ahead of time to ensure that they've got a seat for you, but it isn't a huge hassle usually.
Tips:
FIGHT for a Cargo seat if they din't automatically offer you one. And get the window seat. I say this just because you can put him closest to the window and yourself on the other side, which makes a nice (if small) "play area" on the floor, with you, the corner, and the seat making the "walls" of the play area. This was the ONLY thing that kept Philip from spinning his head around and chucking pea soup at everyone on the plane.
Bring ziplock bags. You can store wipes, towels, barfy clothes, etc in these without getting the rest of your carry-on stuff icky. They've saved our lives many, many times.