It could be argued that there was only one, since they always refer to "the transporter room" and never specify more than one. On the other hand, we did see a number of different "looks" for the transporter room. One had the big star field graphic on the wall, another had a viewscreen and one of those pyramid shaped viewers (like Spock had on the bridge) one the wall behind the console, and another had food slots on the wall near the door. Now, of course, there was only one transporter room set which was simply re-dressed, but, unless you assume the transporter room has some unexpected reconfiguration ability, I'm willing to take it as evidence of multiple, differently appointed transporter rooms. Especially since IIRC more than one of these configuration has been seen in a single episode (though i can't recall exactly which one --- I may be wrong on this point altogether).
I'm working on my own set of deck plans for Kirk's Enterprise (though I don't want to post anything till it's more or less finished) but for my money, I'm using a similar configuration to FJ's tech Manual in that there are four 6-man units, three 22-man emergency evac units, and Cargo transporters. I think FJ only had one of these but I was thinking about having two. (However, I'm putting them in different places than he did.)
There are a few references from the show that should be considered as well. Off the top of my head, "The Enemy Within" deals with the transporter quite a bit. The yellow ore that screws things up in the first place seems to be physically contaminating the system. I interpret this (by shoehorning some TNG Tech Manual stuff) to mean that instead of a each transporter unit having a dedicated pattern buffer, that ALL the ship's transporters share one unit. This has the added benefit of explaining why they always said "the transporter room:" there would only be one transporter room operational at any time since there was only one pattern buffer! The command crew and anyone else who would need to know would know which room was the active room at any given time. This episode also shows that there is a single object in an engineering area that, when shot with a phaser, can cut power to all the transporters.
As to cargo transporters, we have "Dagger of the Mind" In which they are beaming up cargo with the regular 6-man unit. Why would they do this if there was a regular cargo transporter? Perhaps the cargo transporter is such a power hog that they use the more efficient 6-man unit when the cargo size allows for it. Especially if there is normally only one transporter operational at a time; why redirect power from to the cargo unit when it's not really necessary?
Of course, this is just MY interpretation of what we saw on screen. Your milage may vary.