Q: Should your chairs, couches, and bed be on or off your rug, or is a hybrid approach ok?
A hybrid approach is ok. The one thing I don't like with rugs, is for the rug to go all the way to the wall. It's important to always see some floor--otherwise you veer into wall-to-wall carpeting territory.
I like to think of a rug like lipstick: you're putting it on to accentuate what you already have--you're not doing it to become something else. So if, for example, you have hardwood or stone floors, you still want to see bits and pieces of that--you don't want to cover it up completely.
Here are two key tips on placement:
If all of your furniture in the living room is sitting on a rug, that will make the room bigger, and create a sense of grandness. Of course, it depends on how big your room is, so if your rug won't stretch to fit all of your furniture, it's fine to have the front of your couch on it and the back of it off of it, just make sure that your chairs are firmly off or on (both for looks and stability).
In a bedroom, I prefer a rug positioned at the foot of the bed, and a side table sitting off of it. Again, this reasoning has to do with wanting to see the floor--and also an unexplained grudge against night tables on rugs. (Because some things in design don't have a rational explanation, they just are.)
Photograph: Courtesy Derek Swalwell