Hi @Kentishgirl,
Welcome to the Dyson Community, and congratulations on your new Dyson Airstrait™ straightener!
There shouldn’t be any problem with you storing your straightener in either your cabin or hold luggage. You may want to double-check with your airline for peace of mind, but as it’s mains-powered and doesn’t include a battery, we wouldn’t expect there to be an issue here.
Depending on your travel destination, please keep in mind that your Dyson Airstrait™ straightener is engineered to run on the voltage of the country you purchased it in. It won't work at different voltages, and we’d advise against using it with an adaptor - we can’t guarantee this won’t cause damage to your machine. You can find some more information on this here:
Hope this helps, but let us know if you have any other questions!