I realise that my popcorn statement is likely not true, hence why I added "I heard" and a smiley. Perhaps you did not notice the sarcasm.
I noticed it. I was simply adding to it!
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Okay, I will now try to offer some serious input to this topic (god help me)...
One thing I've noticed in my experience with dreams is that the content of my dreams tends to derive from unfinished thoughts earlier in the day, or even week. I've read things that would suggest some truth to this. For instance, say you're alone thinking about something interesting when a friend comes in and interrupts you in mid-thought. You may not get back to that thought process again that day, but your brain still anticipates the continuation of it and may use dreaming as an opportunity to expand on those thoughts.
Undoubtedly of course, the dream(s) is likely to be much stranger than the original thoughts were, and it may deviate so much from your thoughts that you may not recognize any correlation upon waking from the dream.
I have also read that thinking about a particular subject before sleep does not make you dream about it. But I think Cog is correct that simply telling yourself you'll remember whatever you dream makes you more likely to recall it.
If I wanted to sleep during the day and be up at night, would this affect the dreams?
I think it might actually. I have read that differences in light patterns can affect dreams quite a bit. For example, apparently flickering light can cause scary dreams. We are naturally inclined towards sleeping at night, so your brain might interpret too much sunlight during sleep as unpleasant, which may affect your mental state during a dream.
Also, I've always been more likely to be awakened by dreams during daylight hours, even if I didn't need to get up anytime soon.