when I first got one of my bettas I came in one night to find him just lying on his side on the bottom of the tank - I panicked at first because I thought he was sick or dead but he started swimming around normally again when I touched him..
I was still worried about him but then I actually watched him while he was falling asleep one night - he started off resting upright on the bottom of the tank under his cave, and then after a while he just slowly tilted over until he was lying completely on his side. it's pretty funny to watch - especially if something wakes him up suddenly and he has to get himself orientated again.

My other betta seems to sleep upright (at least whenever I see him, anyway)