I have an iPhone 4 (with iOS 6.x on it) that is not synced with any computer.
How can I copy MP3s from a PC to my iPhone? iTunes is not synced with the phone so I cannot put the MP3s into the library and sync.
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If you wish to listed to your MP3s in the Apple's built in Music app for iOS, syncing via iTunes on desktop is your only resort.
If you subscribe to the paid Apple Music service, you can turn on iCloud Music Library and get your MP3s in your iPhone without syncing it with iTunes on desktop. (However, from your MP3 collection, only the songs matching with the ones available in iTunes Store will be made available).
For more information, you can refer this Apple support document, Turn on iCloud Music Library.
The Music app needs to sync with iTunes on desktop. You can use any of the file manager apps for iOS like Documents, transfer the MP3 files to your iPhone. The only limitation being that you'll be restricted to listening to them in the same (file manager) app and not with the Apple's Music app.