Electricity-driven air-source heat pumps are a promising element of the transition to lower-carbon energy systems. In this work, operational optimisation is performed of an air-source heat pump system aimed at providing space heating and domestic hot water to a single-family dwelling. The novelty of this work lies in the development of comprehensive thermal network models of two different system configurations: (i) a standard configuration of a heat pump system coupled to a hot-water cylinder; and (ii) an advanced configuration of a heat pump system coupled to two phase-change material thermal stores. Three different objective functions (operational cost, coefficient of performance, and self-sufficiency from a locally installed solar-PV system) are investigated and the proposed mixed-integer, non-linear optimisation problems are solved by employing a genetic algorithm. Simulations are conducted at two carefully selected European locations with different climate characteristics (Oban in Scotland, UK, and Munich in Southern Germany) over four seasons represented by typical weather weeks. Comparison of key results against a conventional operating strategy reveals that the use of smart operational strategies for the operation of the heat pump and thermal stores can lead to considerable economic savings for consumers and significant performance improvements over the system lifetime. Optimising the operation of the standard configuration leads to average annual cost savings of up to 22% and 20% at the UK and German locations, respectively. The optimisation of the advanced configuration with the two PCM stores shows even higher potential for economic savings – up to 39% and 29% per year at the respective locations – as this configuration allows for greater operational flexibility, and high-electricity-price periods can be almost completely avoided. Depending on the objective function, configuration and location, the system seasonal coefficient of performance varies between 2.4 and 2.8. Lastly, a significant (up to four-times) increase in the fraction of heat pump energy demand covered by an appropriately-sized rooftop PV system is demonstrated, increasing from 8% to 34% at the UK location and from 6% to 24% at the German location. The analysis highlights trade-offs between the objective functions, while the time-resolved results can be used to guide the future development of smart controllers for these applications.