Payment channel networks, such as Bitcoin's Lightning Network, promise to improve the scalability of blockchain systems by processing the majority of transactions off-chain. Due to the design, the positioning of nodes in the network topology is a highly influential factor regarding the experienced performance, costs, and fee revenue of network participants. As a consequence, today's Lightning Network is built around a small number of highly-connected hubs. Recent literature shows the centralizing tendencies to be incentive-compatible and at the same time detrimental to security and privacy. The choice of attachment strategies therefore becomes a crucial factor for the future of such systems. In this paper, we provide an empirical study on the (local and global) impact of various attachment strategies for payment channel networks. To this end, we introduce candidate strategies from the field of graph theory and analyze them with respect to their computational complexity as well as their repercussions for end users and service providers. Moreover, we evaluate their long-term impact on the network topology.