Droplets on fibers have been extensively studied in the recent years. Although the equilibrium shapes of simple droplets on fibers are well established, the situation becomes more complex for compound fluidic systems. Through experimental and numerical investigations, we show herein that compound droplets can be formed on fibers and that they adopt specific geometries. We focus on the various contact lines formed at the meeting of the different phases and we study their equilibrium state. It appears that, depending on the surface tensions, the triple contact lines can remain separate or merge together and form quadruple lines. The nature of the contact lines influences the behavior of the compound droplets on fibers. Indeed, both experimental and numerical results show that, during the detachment process, depending on whether the contact lines are triple or quadruple, the characteristic length is the inner droplet radius or the fiber radius.