It hasn't happened to me yet--or maybe just the usual sort of thing, like an alarm clock incorporated into the end of a dream.
I am fairly aware of what goes on while I'm asleep; but generally this is only a vague sense of where my waking body is, perhaps the level of light in a room, or sounds not loud enough to wake me. These things don't enter into the dream, however.
I do have my own interesting brand of dream, though.
Three times now, an odd but not at all frightening thing has happened to me: I've been going to sleep, and feeling myself gradually lose awareness of my waking body. Generally I first lose the sensation of being in bed, and feel as though I'm floating, and I know I'm going to sleep. Then I lose the sensation of having a body at all, and my field of vision appears, though it's dark--not as though I have my eyes closed. Then, as my vision clears, like fog rolling away, I find myself inside a dream, with a dream-body. At first, I feel tired and sluggish, and the world is indistinct; but then I have more energy and I'm able to move around inside my dream world. I don't think as clearly as I do when I'm awake; but I'm aware I'm dreaming and, if I can imagine something, I can make it happen. I'm particularly fond of, and adept with, telekinesis; but I'm learning how to create objects and change my surroundings too.
I have lucid dreams (with varying levels of control) quite often. It's really a fascinating talent (though it does come with a downside: Lucid dreams are an indicator of very light sleep), and quite entertaining. Usually, however, I realize I'm dreaming in the middle of a dream, after having gone to sleep normally; though I do get the "tired, sluggish" feeling, and a sensation of the world graying out or going indistint, when I know I'm going to wake up. These few incidents in which, apparently, I never quite "lost consciousness" at all while going to sleep, make me wonder exactly what else I can do with my dreams...