We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations where all the defining relations are of the form r = 1. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results to not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented. We also apply these results to establish a close connection between the question of finite presentability of the groups of units of special one-relator inverse monoids, and the open problem of whether all one-relator groups are coherent. In addition, we give a new approach via regular languages for computing invertible pieces of relators in special inverse monoids. We prove that this algorithm outperforms the classical Adjan overlap algorithm when applied to special inverse monoids, in the sense that it discovers more invertible pieces of relators.