I don't sleep deeply on planes. And when I do nod off, I'm in that state where I think I'm awake but I'm actually dreaming.
Usually then I start dreaming that I'm in a plane narrowly avoiding a crash. Often we are taking off from a narrow and short runway, and have to rotate more steeply than normal to avoid the traffic lights and telephone wires. Sometimes we can't get the altitude and have to fly under them and try to cimb before we hit the next batch. Sometimes there are other plance parked there. They're not in the wrong place -- it's just a really crowded airport. Regardless of where I'm flying to/from those take off dreams usually take place at JFK.
I also have similar dreams on landing. Usually in the dream, we're on approach to SEA (though sometiems the dream is about BFI). For some reason, though, we are landing on the street and, again, those traffic lights get in the way.
Most of the time we make it. Or I "wake up" and realize it was only a dream, only to realize in a little while I'm still dreaming. I've had those recursive loops go 5 or 6 levels deep, which sucks.
And then there were the various post-9/11 dreams which stopped sometime around 2003. But that's another story.
Cromely.