Aims: To understand the formation and evolution of the different stellar populations within our Galaxy it is essential to combine detailed kinematical and chemical information for large samples of stars. The aim of this work is to explore the chemical abundances of neutron capture elements which are a product of different nucleosynthesis processes taking place in diverse objects in the Galaxy, such as massive stars, asymptotic giant branch (AGB) stars and supernovae (SNe) explosions. Methods: We derive chemical abundances of Cu, Zn, Sr, Y, Zr, Ba, Ce, Nd, and Eu for a large sample of more than 1000 FGK dwarf stars with high-resolution (R 115 000) and high-quality spectra from the HARPS-GTO program. The abundances are derived by a standard local thermodynamic equilibrium (LTE) analysis using measured equivalent widths (EWs) injected to the code MOOG and a grid of Kurucz ATLAS9 atmospheres. Results: We find that thick disc stars are chemically disjunct for Zn and Eu and also show on average higher Zr but lower Ba and Y than the thin disc stars. We also discovered that the previously identified high-α metal-rich population is also enhanced in Cu, Zn, Nd, and Eu with respect to the thin disc but presents lower Ba and Y abundances on average, following the trend of thick disc stars towards higher metallities and further supporting the different chemical composition of this population. By making a qualitative comparison of O (pure α), Mg, Eu (pure r-process), and s-process elements we can distinguish between the contribution of the more massive stars (SNe II for α and r-process elements) and the lower mass stars (AGBs) whose contribution to the enrichment of the Galaxy is delayed, due to their longer lifetimes. The ratio of heavy-s to light-s elements of thin disc stars presents the expected behaviour (increasing towards lower metallicities) and can be explained by a major contribution of low-mass AGB stars for s-process production at disc metallicities. However, the opposite trend found for thick disc stars suggests that intermediate-mass AGB stars play an important role in the enrichment of the gas from where these stars formed. Previous works in the literature also point to a possible primary production of light-s elements at low metallicities to explain this trend. Finally, we also find an enhancement of light-s elements in the thin disc at super-solar metallicities which could be caused by the contribution of metal-rich AGB stars. Conclusions: This work proves the utility of homogeneous and high-quality data of modest sample sizes. We find some interesting trends that might help to differentiate thin and thick disc population (such as [Zn/Fe] and [Eu/Fe] ratios) and that can also provide useful constraints for Galactic chemical evolution models of the different populations in the Galaxy. Based on observations collected at the La Silla Observatory, ESO (Chile), with the HARPS spectrograph at the 3.6 m ESO telescope (ESO runs ID 72.C—0488, 082.C—0212, and 085.C—0063).Full Tables 1 and 3 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/606/A94