The International Civil Aviation Organization (ICAO) doesn’t allow you to store your e-cigarettes, and most specifically your batteries in your checked baggage in the freight compartment. However, e-cigs are allowed in your carry-on luggage.
E-liquids containers can be stored either in your checked baggage or in your carry-on items as long as they do not exceed 100ml. They also must be put in little see-through plastic bags so that they can be controlled by security agents.
As for the use of electronic cigarettes during flights, it is forbidden due to the fact that the vapor created by e-cigs can potentially trigger smoke detectors. Charging your e-cig is also forbidden.
Last little piece of advice. Pressurization and depressurization of the cabin can generate leaks on e-liquid containers, even if they are new! Don’t hesitate to empty your clearomizers before your flight. And to carry your e-liquids in sealed containers such as plastic bags and boxes for instance.
In trains, buses or subway
Contrary to airplanes, there are no constraints regarding the storage of your material. However we would advise you to turn off your e-cigarette and to pay attention to the state of your batteries. If they are damaged and they come in contact with coins, keys or any other metallic element, this could generate an unintentional de-gassing.
Moreover, it is forbidden to vape in trains, closed buses or subways. In open buses and when there are no explicit ban, you can vape. As for train stations, metro platforms or bus shelters, you’ll need to pay attention to warning signs or posts to check if a local regulation bans vaping.