12th Atrean Dragoons.
The Atrean Dragoons in general get to cherry-pick from the FWL's academies, so they get some of the best and brightest. We know that the Dragoons internal traditions don't favor newcomers and so most soldiers serve less than 5 years in the unit. I've long translated this to mean the unit works a little like a biker gang or a fraternity. If you do the time and put forth the energy, you become truly part of the club for life and get some perks that the new kids don't. For some reason this appeals to me.
Some of the FWLM's best (or at least most advertised) leaders have come through the 12th. Julian Langsdorf, Kelian Brackey (later a Republic Paladin), Christian Hampton (he was sleeping with Alys Rousset-Marik and the father of her children).
The unit was featured in the novel Price of Glory, involved in that now infamous part of history that led to the discovery of the Helm memory core by the Gray Death Legion. I have not read this novel yet (I just ordered it off Amazon actually...). So I'm not sure if reading it will change my views on the unit or not, but I remain optimistic.
Despite being one of the most active and prestigious units, the 12th has a checkered history. They fought on the losing side in Anton's Revolt and the decision to rebuild them was highly controversial. That adds some flavor and dimension to their long history.
The 12th have a cool paint scheme, a cool crossed sabers crest that I can imagine hanging over the fireplace at some regimental headquarters or museum, and are definitely one of the most active units in the FWLM. Even during the relatively quiet years of 3045-3066, the unit saw action in the late 3040s on the eve of the Clan invasion, and was heavily involved in one of the most dramatic battles of Operation Guerrero. Later they played a big role in the Jihad, liberating Sandhurst on Terra from the Blakists. The 12th Atrean Dragoons definitely seem to be one one of the "favorite" units of the god-like writers who craft the canon storyline.