The transition towards more sustainable, fossil-free energy systems is interlinked with a high penetration of stochastic renewables, such as wind and solar. In this context, energy storage technologies of different kinds - namely electrical, thermal and chemical - are commonly expected to play a major role; however, this role is seldom quantified using whole energy system models. In this paper, our goal is to assess and quantify the role of these different storage technologies in low carbon energy systems. To do this, we apply EnergyScope TD, a novel open-source energy planning model, to the real case study of the national energy system of Switzerland; concretely, we optimise the Swiss energy system for a target future year with an hourly resolution. The results indicate the following trends: (i) Thermal storage in combination with heat pumping becomes the main source of flexibility in the electricity sector; (ii) electrification allows a high penetration of renewables in the transportation sector, and the remaining share of mobility is supplied by synthetic fuels; (iii) a mix of storage technologies is needed for different applications at different timescales, such as synthetic fuels for long-term and for mobility demand, and thermal storage for short term (day-week) and for heat demand. Overall, it emerges that storage technologies - such as thermal storage, vehicle to grid and synthetic fuel storage – can help decrease cost, decarbonise the mobility and heating sectors, and reduce dependency on fossil fuels.