The nickel.
13 letters
...Of what US coin? There have been several US coins intended for circulation ranging from half a cent to $20.
If the coin has a mintmark it is under the eagle and above the letters DO in dollar on the reverse of the coin. But not all US coins have mintmarks, if it has none the coin was made in Philadelphia.
No US states have a capital with only 4 letters; they all have 5 letters or more.
North Carolina and South Carolina have 13 letters in their name. Massachusetts has 13 letters in the name.
Nickel is an element that has the same name as a US coin worth five cents.
The US singer you are looking for is Bruno Mars. His first name, "Bruno," has five letters, and his surname, "Mars," has four letters, fitting the criteria provided.
A quarter
the nickel.
dime
Six states have five or less letters in their name. Iowa, Ohio and Utah have four letters in their name. Idaho, Maine and Texas have five letters in their name.
If the coin has a mintmark, it's on the reverse above the letters DO in dollar.