The local opening of DNA is an intriguing phenomenon from a statistical-physics point of view, but is also essential for its biological function. For instance, the transcription and replication of our genetic code cannot take place without the unwinding of the DNA double helix. Although these biological processes are driven by proteins, there might well be a relation between these biological openings and the spontaneous bubble formation due to thermal fluctuations. Mesoscopic models, like the Peyrard-Bishop-Dauxois (PBD) model, have fairly accurately reproduced some experimental denaturation curves and the sharp phase transition in the thermodynamic limit. It is, hence, tempting to see whether these models could be used to predict the biological activity of DNA. In a previous study, we introduced a method that allows to obtain very accurate results on this subject, which showed that some previous claims in this direction, based on molecular-dynamics studies, were premature. This could either imply that the present PBD model should be improved or that biological activity can only be predicted in a more complex framework that involves interactions with proteins and super helical stresses. In this article, we give a detailed description of the statistical method introduced before. Moreover, for several DNA sequences, we give a thorough analysis of the bubble-statistics as a function of position and bubble size and the so-called l-denaturation curves that can be measured experimentally. These show that some important experimental observations are missing in the present model. We discuss how the present model could be improved.