I will allow everyone else to address WWE, WCW, and TNA for this particular topic; I'll instead be making numerous ROH contributions.
Firstly, the Group of 2003 led by Steve Corino. Many will first point to the faction's uncreative name and have a giggle, but that was the least of its problems. While the individual members were solid with new sensation Samoa Joe, terrific heel Michael Shane (ROH's best before CM Punk's feud against Raven), and ECW alumnus CW Anderson, there was no real substantial purpose behind this faction. Forget that Corino had other commitments such as Zero-One at this point in his career and Xavier had a lengthy absence due to a concussion; who the hell was aching to see Corino, Joe, Shane, & Anderson feud against the Prophecy of Dan Maff, Xavier, Donovan Morgan, & Christopher Daniels? It was tedious from inception and the entire idea should've been aborted on the scrapping room floor. Gabe Sapolsky and ROH ought to be highly thankful for the Raven vs. Punk and Corino vs. Homicide feuds boiling in 2003, otherwise this Group vs. Prophecy angle would've had many questioning Sapolsky's storytelling abilities.
The Embassy 2009-2014 were an absolutely useless faction that provided the most minimal value even for match quality. At its best, it was harmless for members; at its worst, it stalled momentum for many. As soon as Prince Nana left the company in 2006, there was never any reason to resurrect the faction; just leave the tremendous memories alone from the feuds against CM Punk, AJ Styles, and Generation Next, and find something fresh for Nana to do when he makes his various returns since 2009, especially since the company couldn't be bothered to have Nana becomes the Kings of Wrestling's mouthpiece at the expense of Shane Hagadorn.
Another lousy Sapolsky idea was in 2007 when he went in a faction warfare direction; this was due to the success that company partner Dragon Gate had with it. By focusing so much on what were mostly useless factions, the product suffered overall especially with lack of detail for Nigel McGuinness's chase to the top. Such ROH factions formed in 2007 include:
The Resilience - led by Austin Aries, other members included dry personalities Matt Cross & Erick Stevens
Hangmen Three - led by often ennui-inducing Adam Pearce, other members included dry personalities Brent Albright and BJ Whitmer
Vulture Squad - led by Jack Evans, other members Ruckus and Jigsaw, with Julius Smokes as their mouthpiece after Homicide's departure. This particular faction has the worst opening sound effect I've ever heard for a wrestling theme; fortunately for you, I'm unable to find it online to share.
The faction warfare fell apart for a number of reasons by 2008, never resulting in anything monumental. Ultimately though, it was something that should've been aborted on the scrapping room floor; even if better decisions had been made with it, such as keeping the Kings of Wrestling together as part of Larry Sweeney's Sweet 'N Sour Inc and Aries recruiting newcomers Kevin Steen & El Generico in his war against Roderick Strong's No Remorse Corps. The result was a year that showed a STEEP decline in Sapolsky's storytelling performance and many on the ROH roster suffered for it with lousy or even no directions, such as McGuinness as already mentioned, as well as Steen & Generico after their feud ended against the Briscoes.