THE VARIETIES OF THE ROTARY STEAM ENGINE.
While there may seem to be a confusing number of types of rotary engine, they can be divided into relatively few categories. Here is my attempt:

Please note that this section is in course of arrangement and does not currently include all the rotary steam engines in the gallery.

THE DIFFICULTIES OF THE ROTARY STEAM ENGINE.
I am no steam engineer (as will no doubt become apparent) but let's see if we can work out why the problems were so intractable. The fact that today there is virtually no market for such a device will be ignored.
The intriguing thing about this particular technological dead-end is that at first sight there seems to be nothing inherent in the concept that makes it so completely impractical. And yet, literally hundreds of engineers and inventors applied their intelligence to the problem, but none were able to make it work acceptably.

Contemporary comment makes it clear that sealing was a major problem; foreshadowing the later sealing problems with the Wankel. Sealing at both the rotor peripheries and the ends was difficult; anything approaching steam tightness meant excessive friction, and either way the machine was inefficient.
Inevitably the thought arises that given modern seal materials and technology, it ought to be possible to make it work. I am no expert on rotary IC engines either but I understand that the sealing problems of the Wankel IC engine have finally been solved, though the low efficiency due to the long thin combustion volume remains an inherent drawback. Surely keeping a rotary machine steam-tight should be much easier, given that the steam will be both cooler and at a lower pressure?

In many cases the rotary inventors seem to have been relying purely on close tolerances and accurate machining to control steam leakage. This is fine in principle, and when appropriately applied (eg the non-contact labyrinth seals used in steam turbines) but much less effective in the topology of a rotary engine, especially when friction and thermal expansion are considered.
It seems inevitable that the inner rotor will get hotter than the external casing, so a rotor that is a snug fit when the machine has just been started will get tighter and tighter as it warms up. A further complication is that unequal heating and thus expansion of each part will result in changes of shape as well as size that will make attempts at steam-tightness through tight clearances quite hopeless.
Few of the drawings above give any clue as to how the ends of the rotors were to be sealed against the side of the casing; it is not just a matter of making clearances small. The thermal expansion of rotor and casing are unlikely to be the same in all directions, and in any case it is equally unlikely that the various components will reach the same temperature at the same time, especially under varying loads.

Contemporary comment also frequently refers to excessive frictional losses due to packing being over-tightened in attempts to obtain acceptable sealing.

Another problem was that many of the designs seem to make no provision for the expansive use of steam, which would have made them very inefficient compared with normal reciprocating engines, even if the sealing had been frictionless and perfect: expensive rather than expansive. Likewise, many designers seem to have had no notion of getting steam into and out of the working chamber easily- the steam passages were often narrow and convoluted.

Now consider the reciprocating engine and its piston. ("piston" is derived from the French for "pestle"- beautifully descriptive!) There is movement in one direction only, and the whole situation is mechanically far simpler. Thermal expansion is dealt with by piston rings that press against the bore due to their own elasticity. (and valve sealing does not seem to be a problem: cf using a compression tester on an IC engine)

The essence of the situation is that sealing a reciprocating piston is a one-dimensional problem. With a cylindrical piston the sealing situation is the same all the way round, as it were, and the sliding movement does not alter the geometry. In contrast, the rotary engine presents a three-dimensional sealing problem, usually continuously altering in its geometry. This is quite a different matter.

THE MARKET FOR THE ROTARY STEAM ENGINE.
Well, essentially there isn't one. In the relatively few places where a small steam-engine is required, a steam turbine, usually geared down to its load, will be more efficient, more reliable, and easier to maintain. One example is on board a turbine-propelled ship, where there is plently of steam available but electricity is in relatively short supply. Turbines were often used to drive the fans that pressurised the boiler-room, replacing the small reciprocating engines that first took on this role. On land, no-one today would dream of installing a steam boiler without electrical controls. And if mains electricity is available and already essential for operation, any minor requirements for power, such as driving pumps, might as well be met by an electric motor.

CONCLUSION.
By 1910, the British Patent office alone held over 2000 patents for rotary type pumps and engines. Many promising applications, such as driving electrical generators on steam locomotives, were taken over by small steam turbines, which gave a usefully higher rpm.
However, rotary air compressors and motors have been highly successful; presumably because leakage at the clearances is less of a drawback than for a steam-engine. Note, however, that compressors for very high pressures are invariably of the piston type.

The Rotary steam engine was a seductive concept, but the practical difficulties appear to have been severe, and it languishes in almost total obscurity. No doubt the problems could be overcome given sufficient incentive; consider the slow and painful but ultimately successful efforts to make a durable Wankel engine. Consider also that the piston engine remains near-universal...
Today there are very, very few possible applications for the Rotary steam engine; in almost all cases an electric motor powered from a central power station will beat it on efficiency, convenience, and just about every other consideration you can think of.

SOME BRITISH ROTARY ENGINE PATENTS

Patent No.	TITLE	NAME	DATE	TYPE	VIEW
25,388	Improvements in Rotary steam Engines	Robert Keats	1901		View patent
26,452	An Improved Engine with Rotary Piston	Thomas Kemp	1901		View patent
18,513	Improvements in Rotary steam Engines	Benson	1914
194,409	Rotary steam engine	Arthur Trotter	1923
200,155	An Improved Rotary steam engine	Alexander Hall	1923
280,267	Rotary steam engine	James Bonham	1927
877,500	A Rotary piston steam engine	Freethy Champion	1959

Mr Champion's design is particularly noteworthy as it uses a nuclear reactor to raise steam!

Some Canadian patents: