Good timing on the question, as we are coming into the colder months when heaters come on and the air gets drier. For the health of the guitar, it likes to be in a relatively constant environment. Rapid changes in temperature such as leaving a guitar in a non-running car in the winter can cause a guitar to crack or have other problems immediately. A bit more subtle and slow, is getting an instrument too dry which leads to similar problems. A case helps slow and buffer environmental changes that the instrument experiences. Also a case protects against impact damage. As a result, a case is safer (better) for an instrument than sitting out. However, as Bulldog mentioned, he keeps his house humidified (I do as well). By controlling the temperature and the humidity, the only remaining problem is rash and damage from unintentional contact with the instruments. If you aren’t worried about that, then keeping your instruments out gives you the opportunity to pick them up and play them whenever you want, even if only for a few minutes. Plus, looking at Bulldog’s room, I think they also look good out there where everyone can see them.
As for me, I generally have one or two instruments out and the rest are in their cases. They get rotated around quite a bit, so it’s not like I have instruments that stay out and some that stay put away. Well, that’s not really true… I forgot that my PBass stays out. I lent it to a friend for a few decades, and it went through a flood and then apparently sat against a heater for a bit (it did a nice relic job for me). The case was mildewed, so I hung the PBass out for a while and it has stayed there, but a solid body is a bit more robust than an acoustic.