It's worth pointing out that Commodore is only no longer a rank in the US Navy. It's still 1-star flag in the British Royal Navy, several European navies and in the maritime forces of most of the former Commonwealth. Commodore is used as a nick-name for some type of special commanding Captain in the US Navy, I think, but that's all.
Fleet Captain was actually a title-come-rank used in the Royal Navy during the 18th century and in some German Navy at some other time (not sure when). It was also proposed as a real rank between Captain and Rear Admiral during the latter years of WWII in the US Navy (Commodore had been dropped at this point) or to be given to aircraft carrier captains, but was rejected. As a title, it is used in the civilian maritime field as the senior captain of a yachting club, just below Vice-Commodore.
In the entirety of TNG, DS9 and VOY, there were only two supposedly 1-pip flag officers seen, and both were in Season 1: Admiral Mark Jameson ('Too Short A Season') and Admiral Gregory Quinn ('Coming of Age', 'Conspiracy'). However, since the rank insignia wasn't really established for flag officers then, it's kind of unclear as to if they were 1-pins or not. None were ever seen in the later 'boxed' pin style. Noteworthy, the vast majority of flag officers seen in this period were 3-pin Vice Admirals, with only a handful of 2-pin Rear Admirals and one or two 4-pin Admirals. No 5-pin has ever been seen outside of the TWoK-era.
For reference, TOS only really showed us 1-pin Commodores. A handful of Admirals popped up, but only one was seen with rank insignia (a Vice Admiral). The TWoK era used the whole spectrum. ENT showed several 1-pin flag officers. Only one was named (Commodore Forrest, 'First Flight'), but others were seen in the background. The only 1-pin seen was Rear Admiral Leonard in 'Broken Bow' (possibly some in background, I don't know). Vice Admiral Forrest was the only 3-pin. No 4-pin or 5-pin were seen in this era.
Why do I know all this? Military uniforms are an interest of mine, so I tend to look for these things on the actors when they appear
