There is little evidence our army issued or promoted the use of spare cylinders for reloading, likely due to the doctrine of using the handgun to to close and them employing the saber. Most troops did not carry a handgun, they were mostly limited to officers and cavalry troops (hence the saber doctrine)
There was though at least one army that issued the colt 1851 with a spare cylinder and a holster pouch combination as well are sperate cylinder pouches and that was the large and well known and well-respected Prussian Army, as evidenced in my earlier post in this thread.
That we see little evidence of it here is due to that fact that handguns themselves were not "general issue" and confined to certain types of soldiers, mostly officers and cavalry. Calvary likely didn't make use of the spare cylinders for two reasons, the aforementioned use of the saber in close quarters and, as they were on a horse, the ability to carry more weight in the form of extra revolvers. They even had those pommel holsters designed to carry extra larger heavier revolvers. And officers, well likely reloads were not a priority for most of them for the obvious reasons and if they were in combat, they also carried sabers.
You would also, in the case of the Confederacy have to take into account the general shortage of guns, and in particular revolvers, which would have likely driven the use of spare cylinders as getting one revolver was difficult and multiples more so, so the cylinder thing would have been more likely.
The only thing that there is little evidence of is pouches, that spare cylinders were sold and even advertised is well documented as well as pistols coming with spare cylinders. On thing I tend to dismiss those naysayers that right about this issue is, when they start citing the dangers of carrying a "loaded" cylinder, comparing them to hand grenades or claymore mines. Yes, a capped loaded cylinder would present hazards, but a loaded uncapped cylinder would not as we all here should be well aware of, and these "expert" writers more often than not do not differentiate or even mention this. Even an uncapped loaded cylinder would greatly sped up the reloading process, especially if on had a capper around their neck. Uncapped cylinders would also be easier/safer to carry, thus not requiring a special pouch, explaining their scarcity.
A good question may be did those carry spare loaded cylinders carry them capped, or uncapped?
As far as commonly done, I would say the Prussian army's use pretty much says it was "common". Depends on how you define common though as it is a subjective concept.