From what I've seen on Aether...

Flames = Sargatanas
Storm = Balmung
Adders = Faerie

I'm sure there's other servers with popular linkshells, but those are the ones I'm aware of. One thing I'll mention about Faerie which probably applies to others, while we often do have people in our linkshell running games, there's a lot of times people do other stuff, aren't online, have full parties or only get a few people going in together, so it's definitely not perfect but if you don't have huge expectations of non-stop games it's pretty good.