These are powers in the Constitution given to the national government.
Power to lay and collect taxes
Power to regulate commerce
Power to establish post offices
Power to establish a rule of naturalization
Power to create courts inferior to the Supreme Court
Power to declare war
Power to raise an army and Navy
Power to make laws which are necessary and proper (Necessary and proper clause)
Powers given to both the national government and the states.
Power to tax
Power to build roads
Power to establish bankruptcy laws
Power to create lower courts
This stems from the 10th amendment, which says "The powers not delegated to the US by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people."
Power to regulate trade within the state
Power to conduct elections
Power to establish local governments
Power to establish public school systems
Powers not listed in the Constitution that stem from the aforementioned Necessary and Proper Clause.
Examples include creating a national bank, or regulating highways.