I'm in the same boat--the term "morningstar" is a confusing one, especially when it's inconsistently used. For game purposes, it's the bludgeoning & piercing spiked club, though it reality, it can easily be something different.
A show on the History Channel, Conquest, has covered various medieval weapons in some episodes (in focuses on winning in various situations, from sports to combat), & it gives you a decent, basic idea of things.
One of the thing with one-handed flails is that they were supposed to be in constant motion--it's the momentum of the whirling ball (or balls, since some flails did have multiple heads) that does the damage. Not giving it a chance to build up momentum lessens the impact of the weapon. However, there was always a decent chance of the wielder being struck by the flail (thus a double-ended flail is much more hazardous & impractical, IMHO--momentum would have to be maintained for both flail heads, and the range of movement would be rather limited in order to prevent hitting yourself with either/both of the whilrling flail heads--but that's another topic altogether).
As for how a flail could be carried--well, I'd say it depends on the size of the flail. Heavy/two-handed flails probably would be carried around like a polearm, large axe, or other weapon with a long shaft/haft. A smaller, light/one-handed flail would best work stowed in an open side pouch/bag, large enough to hold the ball and chain, as well as support a good part of the handle, but small enough to allow part of the handle to stick out of the bag--basically enough to allow a good hand hold when grabbed quickly. It's feasible to secure the weapon by sealing the bag & binding the handle to the bag or belt (but this pretty much eliminates any quick-draw possibilities).
When the weapon is just stowed away, I'd guess it'd be in a large enough container, with the chain wrapped around the handle if possible, and with the head wrapped in cloth or some other padding material if the ball is spiked.
Reasonably, if a weapon is meant to be accessed quickly, then it should have a container that is quick & easy to draw the weapon from. A belt-ring works great for handaxes, maces, hammers, & other "fixed" hafted weapons, but not a flail (you'd have to thread the handle & chain through the ring constantly). A scabbard is great for a sword or knife. However, "odd" weapons like a flail probably would be stashed in an open bag on a belt to quickly ready the weapon.
The only other option for a one-handed flail is to fold the ball and chain against the handle, & tuck it in a belt. The end of the handle and the ball would be above, and the rest of the chain, handle, and hinge would be below.