Recent developments in the electric power sector as well as in district heating and cooling systems has led to an increased interest in local energy systems and markets. In the electricity sector, this is driven by the integration of distributed resources such as solar power, electric vehicles and demand response. For district heating, sustainability and energy efficiency targets drives the development to further exploit small-scale heat sources. A closer integration of these energy carriers can also unlock potential flexibility, to the benefit of local as well as overlaying systems. In this respect, there is a need to further explore the possibilities to design local energy markets to facilitate the integration between electricity and district heating, as well as providing adequate instruments enabling flexibility. This paper therefore presents a market clearing design, based on optimization, for local energy markets incorporating multiple energy carriers and bid structures suitable for representing flexibility. The market clearing model is applied in a case study to illustrate and validate key design elements. One conclusion is that even though various elements can be added to the market clearing function, there is a challenge to interpret the results due to an increased complexity of the resulting optimization problem.