We study the behaviour of solutions of analytic differential equations from the point of vue of o-minimality. It is well known that the structure generated by the non spiraling leaves of codimension 1 analytic foliations is o-minimal. We investigate the properties of non oscillating trajectories of analytic vector fields. We show that, under some sufficient conditions related to the notion of quasi-analyticity, these trajectories belong to o-minimal structures. We also give some examples of non oscillating trajectories which do not belong to any o-minimal structure, and examples of infinite families of o-minimal structures such that any two of them do not admit an o-minimal common extension.