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" music. The moment someone creates a creative work, 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 on a piano, records it and puts it on the Internet, they might not have a copyright on the melody, but they have a copyright on this specific recording. That means not just the song needs to be old enough to be in the public domain, but also the performance. Any recordings taken over 70 years ago will likely be of insufficient quality for your game, so they are likely out of discussion.
- Music where the copyright holder(s) offer(s) the work under a free license. In that case check the license conditions of the piece in question to see if you are allowed to use it for your specific purpose and if you need to fulfill any conditions in order to do that. Some musicians release their work under a license which allows non-commercial use but not commercial use (with various definitions of "commercial"). Some might allow commercial use, but only if you credit them and/or do not edit their work. A few might allow you to do anything you want. But unless you actually see a license attached to the work, you have to assume that it's an "all rights reserved" work which must not be used without permission.
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 search the web for royalty free music composers. 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.