Recently, we had an incident in the cabin during boarding (I work for an airline) which lead to deplaning all the passengers. I was one of the team members who were assigned to investigate the incident.
The passenger was following the policy, which is almost the same as mentioned in the other answer by @JoErNanO. Unfortunately, the batteries just explode, causing a severe smoke and it actually ignited. Luckily the crew put it off in a matter of seconds.
Imagine if this happened after the service when all lights are off and most passengers are sleeping. It would have taken more than a few seconds to identify the source of the smoke and that could give the fire a chance to spread. What about if this happened in the cargo hold!
Even though there was no injuries caused by that incident, it really had the potential to be a serious one. Hence, many airlines including the one I work for started a campaign to educate passengers about how dangerous they are. The LiPo bags are the best solution, as they will contain the smoke/fire within.
Regarding airport security, some airports already implemented strict policies regarding this, they basically won't allow any mid-large LiPo batteries to be carried by passengers, especially power banks as they have large batteries. China in particular is very strict regarding this. China actually knows that they make bad batteries. I have heard that they allow good Chinese brands. I was told that by a salesman there while I was buying a quadcopter in China recently, I can not confirm this anyway.
One last thing, I have ordered a cheap tablet online. It was delayed for a few weeks. The shipping company called me and informed me that the battery was the reason and many operators have stopped shipping that unless batteries are wrapped in a special way and they are implementing new rules for that.
Bottom line: even though they are still widely allowed as carry-on, they are still dangerous and if you have large batteries with high capacity it is best to put them in LiPo bags. Same exact advice is applicable when you have loose ones at your home like the ones that come with all kinds of RC planes, it is better to keep them in LiPo bags while not using them, last thing you want is a battery to ignite and burn your house down.