Local Energy Markets (LEMs) were recently proposed as a measure to coordinate an increasing amount of distributed energy resources on a distribution grid level. A variety of market models for LEMs are currently being discussed; however, a consistent analysis of various proposed grid tariff designs is missing. We address this gap by formulating a linear optimization-based market matching algorithm capable of modeling a variation of grid tariff designs. A comprehensive simulative study is performed for yearly simulations of a rural, semiurban, and urban grids in Germany, focusing on electric vehicles, heat pumps, battery storage, and photovoltaics in residential and commercial buildings. We compare energy-based grid tariffs with constant, topology-dependent and time-variable cost components and power-based tariffs to a benchmark case. The results show that grid tariffs with power fees show a significantly higher potential for the reduction of peak demand and feed-in (30–64%) than energy fee-based tariffs (8–49%). Additionally, we show that energy-based grid tariffs do not value the flexibility of assets such as electric vehicles compared to inflexible loads. A postprocessing of market results valuing the reduction of power peaks is proposed, enabling a compensation for the usage of asset flexibility.