Squadrons of Galactica
Numerous squadron insignia are visible on the walls of the pilots' ready room on Galactica. The squadrons can contain both Viper and Raptor pilots.
It should be noted that these are purely organizational in nature. In combat, Viper squadrons are color-coded; for example Blue Squadron. These designations could be wireless code names that are assigned on an ad-hoc basis as needed for each missions.
Furthermore, it is likely that not all visible squadrons are active. Given the relatively low number of Vipers on Galactica compared to her standard operational status with two active flightpods, it is possible that some of these squadrons served on Galactica during the battlestar's long history, but have since been assigned elsewhere or disbanded. This could be the case with Primus squadron which served in the first Cylon War and whose patches might have been kept for historical or nostalgic reasons.
Primus
Primus is the name of the 1st Viper Squadron of the Colonial Fleet, although it is unknown if it is still in active service.
William Adama was a member of this fighter group, in addition to other pilots from the Cylon War era such as Raymond Lai. The logo of this group is adorned prominently on the nose of many restored Viper Mark II fighters that were to occupy the Galactica Museum.
A photo of a younger William Adama and his two boys, Zak and Lee, shows the Primus logo on his jumpsuit and his old fighter, Viper 7242.
Vigilantes
The Vigilantes, 3rd Viper Squadron, is a long-established unit in the Colonial Fleet before the Cylon attack. Given that the squadron's logo adorns a wall in the quarters of Colonel Tigh (a commanding officer long removed from pilot service), it is likely that Tigh served as a member of the Vigilantes as a Viper pilot in the first Cylon War.
Captain Kara Thrace is a member of the Vigilantes (Act of Contrition).
Jackson Spencer, the former CAG of Galactica, is also a member of the Vigilantes. He and his squadron were completely destroyed by Cylon Raiders during the Cylon attack on the Twelve Colonies. The only known survivors at this time are Thrace, who was in the brig for striking a superior officer, and the crew of the Raptor supporting the squadron, Sharon Valerii and Karl Agathon. It is unknown if the Raptor crew was part of the squadron or just assigned to support it (Miniseries).
The first class of nuggets fly as members of this squadron (Act of Contrition).
Margaret Edmondson and Hamish McCall, Raptor pilots, are both members of the Vigilantes (Lay Down Your Burdens, Part I).