Abstract We consider maximal globally hyperbolic flat (2+1)-spacetimes with compact space S of genus g>1. For any spacetime M of this type, the length of time that the events have been in existence is M defines a global time, called the cosmological time CT of M, which reveals deep intrinsic properties of spacetime. In particular, the past/future asymptotic states of the cosmological time recover and decouple the linear and the translational parts of the ISO(2,1)-valued holonomy of the flat spacetime. The initial singularity can be interpreted as an isometric action of the fundamental group of S on a suitable real tree. The initial singularity faithfully manifests itself as a lack of smoothness of the embedding of the CT level surfaces into the spacetime M. The cosmological time determines a real analytic curve in the Teichmüller space of Riemann surfaces of genus g, which connects an interior point (associated to the linear part of the holonomy) with a point on Thurston's natural boundary (associated to the initial singularity).