In this work, the notion of a quantum inverse semigroup is introduced as a linearized generalization of inverse semigroups. Beyond the algebra of an inverse semigroup, which is the natural example of a quantum inverse semigroup, several other examples of this new structure are presented in different contexts; those are related to Hopf algebras, weak Hopf algebras, partial actions and Hopf categories. Finally, a generalized notion of local bisections is defined for commutative Hopf algebroids over a commutative base algebra giving rise to new examples of quantum inverse semigroups associated with Hopf algebroids in the same sense that inverse semigroups are related to groupoids.