If elements in a group are unitaries, then inverse semigroups are formed by partial isometries of a space. They were introduced in the '50s independently by Wagner and Preston, and have recently received much attention within not only the C*-community but also the dynamic crowd and the groupoid people. In this talk we will study three of their properties from three different points of view. In particular, we shall introduce how to equip an inverse semigroup with a (suitably) invariant metric, which results in an (extended) metric space that inherits much of the algebraic properties of the original object. From this, we will study how to see amenability, which is an algebraic property, in a coarse manner and in a certain C*-algebra. Likewise, we will relate the exactness of that algebra to the property A of the inverse semigroup and the amenability of a certain action on its boundary. Finally, time permitting, quasi-diagonality issues will also be discussed, with a special focus on applying these techniques to traceless C*-algebras. This is based on several joint works with Pere Ara and Fernando Lledó.