What exactly counts as a creature's space, given that there are a lot of effects that care about an "unoccupied space"? Are a creature's space, its controlled space, and its occupied space all the same thing? What can we say about the shape of the space, and the height of the space? Is the occupied space of a Medium size creature just five feet high, even when the creature is larger than five feet?
\$\begingroup\$ Related: The PHB uses 'unoccupied' 49 times but never defines what an unoccupied space is, as far as I can tell, and neither does the DMG. The context of dimension door, and the figurine of wondrous power, make it clear that a space can be occupied by at least a creature or object, but do not explicitly say that these are the only things that can occupy spaces. See also: Is a space with a hazard or Area of Effect 'occupied'? \$\endgroup\$– KirtMay 8, 2022 at 6:16
\$\begingroup\$ What's the case of "occupied space" that you're concerned about? \$\endgroup\$– Mark WellsMay 8, 2022 at 23:48
\$\begingroup\$ @MarkWells Its the context of many situations (especially with spells) that ask for an unoccupied space. Often in combat this will be close to a creature, and so you need to know which space the creature occupies to determine which space is unoccupied by it (independent of other sources of occupation, that may need separate resolution). \$\endgroup\$– Groody the HobgoblinMay 9, 2022 at 3:49
The main rules for Creature Size in the basic rules and PHB state that
A creature's space is the area in feet that it effectively controls in combat
So we know the "creature's space" and "controlled space" are the same thing.
Since that's the only definition we have of a creature's space, we have to assume that's what "occupied space" means, though objects can occupy space even if they aren't creatures -- but a DM will have to judge that for each specific effect. I would personally allow a Medium character to misty step into a space that's smaller than 5 feet wide, but big enough to squeeze into, for example, or teleport under a table in a prone position without worrying about what their combat space is at that moment.
The shape of the space is pretty clear from the Creature Size table: spaces are square. 5x5 feet for a small or medium creature, 10x10 for a large, and so on.
The short version is the height of a creature's space is not clearly defined, and maybe that makes a lot of sense. Horizontal space is much easier to 'control' as you can move and lunge to control a lot more than your body actually occupies, but your ability to strike vertically in combat depends heavily on your actual stature and, in some cases, body-plan. For example, a halfling with a dagger and a great-axe-wielding orc have the same 5x5 combat space, but obviously the orc is going to have an easier time hitting something flying low overhead. Even more to the point, a rhinoceros is a Large creature with a 10x10 space, but can it really hit anything above it, like at all? Its vertical reach should probably be limited to its actual height.
We have asked questions on the boards several times that try to interrogate 'space', such as Is a space with a hazard or Area of Effect 'occupied'? and Can you fly over a Medium enemy creature in a 10ft tall corridor?, if you want to see more.
1\$\begingroup\$ The DMG also has hexagonal spaces, so it need not be square, methinks. \$\endgroup\$ May 8, 2022 at 15:27
2\$\begingroup\$ Using a grid at all is, technically, an optional rule, whether it's squares or hexes. But hex grid is sort of double-optional, it requires a whole subsection to explain how creature spaces work in hex mode. By the PHB default, spaces are all squares. \$\endgroup\$ May 8, 2022 at 23:26