I'm making a game. It will be sold and not free. Am I allowed to use non-copyrighted music in it?
There is no such thing as "non-copyrighted" creative works. The moment someone creates something, they have a copyright on it. What you could mean is:
- Music in the public domain due to age. But that would mean that the music is at least 70 years old, in some countries longer. Remember that melody, lyrics, arrangement and performance are all separately copyrighted. When someone plays Mozart and records it, they might not have a copyright on the melody, but they have a copyright on this specific recording. So you can pretty much forget about music which is in the public domain due to age, unless you reinterpret the music yourself.
- Music where the copyright holder offers the work under a free license. In that case check the license conditions of the piece in question to see what obligations you need to fulfill to use it.
And by the way: Whether or not your derived work is paid, unpaid, for-profit or non-profit is largely irrelevant for what you can and can not do under the copyright laws of most countries.
I don't understand what you mean to say "non-copyrighted". Mayhap, you were trying to say royalty free music. If yes, then you should try to get royalty free music from CreativeCommons but keep in mind that you are using it for a commercial purpose that's why you would definitely need the CreativeCommons license before using this royalty free music.
On the other hand, you can use the royalty free music of indie composer from https://www.musicscreen.org/. However, you would again need a pro license with the permission of musician to use it for commercial purpose as royalty free music sites only offer music without having the license to students to use it for non-commercial purpose.