Inuyasha is an insomniac. Him being so used to watching out for predators, coupled with the fact that he's so restless, renders sleep close to unattainable. He's also prone to nightmares, often about losing his friends and being alone again. Being around Kagome seems to help on both accounts; she makes him feel safe.
Kagome talks a lot in her sleep. Very little of it makes sense. Her dreams tend to be weird.
Shippo, like many small children, has an easier time falling asleep when he can hear someone else's heartbeat. This is why he is seen curled up next to Kagome so often. He still dreams about his late parents, but they're happy dreams.
If you put something within arm's reach of a sleeping Miroku, it will be cuddled. It's inescapable. He dreams a lot about his Kazaana, but later on dreams of Sango take over.
Sango steals the covers. She tends to have long, narrative-style dreams.
Sesshomaru sleeps a couple of hours a week, at most. He doesn't like it when others catch him napping, and he wouldn't sleep at all if he could help it. Much like a dog, he mostly dreams about hunting.
Rin has night terrors, in which she screams and cries and is generally inconsolable. These scare Sesshomaru more than he would ever admit. Most of her nightmares are about the robbers who killed her family.
Kouhaku is also prone to might terrors. His are about the terrible things he did because of Naraku. He goes on very little sleep for a boy of his age.
Naraku, being a hanyou, needs at least a little sleep, but he ignores it until the last possible moment because he sees it as a sign of weakness. He dreams of when he was Onigumo, and of Kikyo.
Kagura sleeps on the job whenever possible, both out of need and as a sign of disrespect. She dreams of flying.