In my work I come upon divisible abelian groups all the time, especially those of the form $\mathbb Q_p/\mathbb Z_p$, or direct sums of such groups. One frequently uses the fact that they are injective (to conclude that certain sequences of Hom groups, which a priori would only be left exact, are in fact exact on the right as well), and other properties as well; for example, groups of type mentioned above, related to $\mathbb Q_p/\mathbb Z_p$, have non-zero $p$-adic Tate modules, which can often be useful.