I remember some. I have sleep apnea and I remember more when my apnea is worse for some reason. I've had some very bad nightmares where I wake in a sweat. The last trrible one I had was a man that hunted two of us (don't know who the other one was but he was with me) and tied us both up with our legs spread and cut the other one with a chainsaw from the crouch up through his head. Now isn't that bizzare? I woke up in a cold sweat then and couldn't get back to sleep for the rest of the night. I got like 3 hours of sleep that night.
I've also had premonitions in my dreams. There is one that sticks in my head from childhood. I was being pulled up out of something (I thought water) and the guy said, "don't worry, I use to lift octopus for a living". Well, weeks later, I was rescued from a rock fall on a mountain and the guy that rescued me said "don't worry, I use to lift octopus for a living", the exact same phrase in the exact same tone. I had never met this man before. I knew right away this was what I had dreamed weeks earlier. I was 10 or 11 years old. I even said to the guy I had drempt this and he just ignored my comment, in hind-sight he probably thought I was in shock. I probably was in shock but the dream I had weeks earlier was very clear in my mind. It was a dream I pondered more than several times afterwards. I purposly tried to remember it after the dream for some reason. I went over it in my mind several times before the actual incident happened. I have had several other premonitions as a child. I don't have them anymore as an adult. I am convinced of this one premonition that I speak of since I gave it so much thought before the actual occurance. The other premonitions were less tramatic and I question if they were really premonitions or coincidences.
In any case, the far majority of the dreams I remember, don't come true.