Which of the following powers may be exercised by local governments under home rule?

Local governments are granted home rule to have authority over certain matters without requiring explicit permission from the state legislature. One of the key powers associated with home rule is the ability to amend their own charters. This means local governments can establish their own governing structures and procedures, reflecting the unique needs and preferences of their communities. Such flexibility allows them to adapt to local conditions and enhance governance.

The other options do not align with the fundamental concept of home rule. Local governments do not have the power to create state laws, which is a function reserved for the state legislature. Controlling state budgets is also outside the purview of local governments, as budgets are determined and managed at the state level. While local governments may enforce certain state laws, they do so under the authority granted by the state, rather than exercising independent power related to law enforcement. Thus, the ability to amend their own charters under home rule is a distinct and essential aspect of local governance.