Flying with a Fountain Pen
The first, most obvious question is: should you fly with a fountain pen? Well, no. Use an aeroplane, they're much better at it. But let's assume you are using a plane. And you just don't know if you're ok to use a fountain pen on the plane, or how to safely take your fountain pen or pens with you.
Use in Flight
Opinions differ, and this is one of those situations where there's an element of risk - many people use a fountain pen with no special precautions, and they're fine, while others use one once and get covered in ink. If you don't mind using something else while you're flying the safest option is just to use a ballpoint or a pencil. We've never heard of a pencil leaking mid-flight!
If you really want to use a fountain pen while flying, it's best to have it as full as possible, so there's less space for air to expand inside, and uncap it carefully tip-up, once you're at full altitude - writing with it while the plane climbs is much more likely to end up with ink all over your notes.
The thing to remember is that you're going to be in lower-pressure air. If there's air inside the pen, sealed in place with ink between it and the outside, it's going to expand. And if that ink is the only thing stopping it, the ink is going to get pushed out of the way so the air can expand.
If you keep the pen tip-up, the air will be at the top, closest to the nib, so there won't be much ink in the way - not too much spilled if it does get pushed out - a bit of tissue will probably save you.
Taking Your Pens
If you don't mind using something else while you're flying, but you do want one or more fountain pens with you, things can be made even safer. Just empty your pens before you go - no ink, no ink to spill. Just in case of any traces, a sealable plastic bag can make things even safer - pop a bit of tissue paper in with them to absorb anything that does leak out.
And if you don't want to empty them and re-fill at the other end? Leave them capped, and in a plastic bag with some tissue, and you should be fine. Carried so they stay tip-up is still safer if you can pack carefully and keep your bag the right way up.
In any case, you either have to accept the small risk of a leak, or take enough precautions to make sure it won't happen - plenty of people fly with fountain pens, and with the most basic care, problems are quite unlikely.