Are there any airlines that allow electronic cigarettes?

I can't give a definitive answer, however I suspect that you will not find any major airlines anywhere in the world that allow electronic cigarettes.

All of the airlines I've flown recently have specifically called our electronic cigarettes as not being allowed on their flights, either in the in-flight magazine, or by specifically stating them as not being allowed during the safety briefing.

Even if you ignore the potential confusion between real and electronic cigarettes, the fact that these devices potentially give of a harmful vapor which then gets passed into the recirculated air within the aircraft would be enough for any major airline to ban then.

This is an interesting social/legal questions which I expect will change in the coming years. One airline that actively sells e-cigarettes on the plane is unsurprisingly Ryan Air (they'd do anything for an extra £/$/€). Interesting article from the BBC about this subject back in 2012 -http://www.bbc.co.uk/news/magazine-18734753

Consider that even if the airline ignored it, it is very unlikely that your fellow passengers would. I would view use of an electronic cigarette as something similar to passing wind, with the exception that passing wind is not always avoidable for biological reasons. I would certainly not consent to someone near me using one in an enclosed space such as inside an aircraft cabin.

Also, in some airspace jurisdictions it is illegal to smoke or release other gases into the cabin of a flying aircraft. You may find the local police waiting to arrest you when the plane lands.

All international carriers that I have flown on, prohibit smoking on board flights and all have added specifically e-cigarettes to the announcement/safety video to explicitly state that these are also forbidden.

The US Department of Transportation prohibits the use of e-cigarettes and in addition, prohibits the charging of these devices on-board - which is also recommended by ICAO:

U.S. Department of Transportation Pipeline and Hazardous Materials Safety Administration today issued an interim final rule (IFR) to prohibit passengers and crewmembers from carrying battery-powered portable electronic smoking devices (e.g. e-cigarettes, e-cigs, e-cigars, e-pipes, personal vaporizers, electronic nicotine delivery systems) in checked baggage and prohibit passengers and crewmembers from charging the devices and/or batteries on board the aircraft.

“We know from recent incidents that e-cigarettes in checked bags can catch fire during transport,” said Transportation Secretary Anthony Foxx. “Fire hazards in flight are particularly dangerous. Banning e-cigarettes from checked bags is a prudent safety measure.”

On January 22, 2015, the Federal Aviation Administration issued a Safety Alert for Operators recommending that air carriers require their passengers to carry e-cigarettes and related devices exclusively in the cabin of the aircraft. Also, on June 9, 2015, the International Civil Aviation Organization (ICAO) published an addendum to the 2015-2016 ICAO Technical Instructions for the Safe Transport of Dangerous Goods by Air prohibiting the carriage of e-cigarettes in checked baggage and restricting the charging of these devices while on board the aircraft.

Passengers may continue to carry e-cigarettes for personal use in carry-on baggage or on their person but may not use them on flights. The Department’s current regulatory ban on smoking of tobacco products on passenger flights includes the use of electronic cigarettes. Nevertheless, to prevent passenger or crewmember confusion, the Department has proposed to amend its existing airline smoking rule to explicitly ban use of electronic cigarettes aboard aircraft.

In effect this means that all US-flagged carriers prohibit smoking of e-cigarettes on board; all carriers prohibit charging of these devices on board; and could also possibly mean that any carrier flying into the US should also abide by these rules.

For Europe, EASA also has a similar safety information bulletin based on the ICAO bulletin which also has the same restrictions.

I should note that these restrictions go towards charging of the device as it has a heated element which is a fire hazard. As far as I understand charging of the device is an integral part of the entire smoking experience - so it would make sense that smoking them would also be forbidden.

Finally, I don't know anyone who would assume vapestick (a manufacturer of e-cigarettes) is the authority on where they should be used. Even if you ignore the bias, it is a very, very, very bad idea to smoke in the toilet of an aircraft. It is the fastest way to get a fine, be banned from the airline, and possibly put in jail.