How do I edit songs that are in the Music app on my iPhone 5? I need to delete some songs that were automatically synced that I don't want.
iOS 7 has changed the face of the Music app somewhat. The biggest thing that was introduced is that now, all of you iTunes store music purchases show up automatically in the Music.app. This allows you to stream those songs without actually storing them on your device (an excellent option).
As a result you have two options:
You can manage music the old way by opening Settings.app -> Music -> and disabling "Show All Music". This will prevent your purchases from displaying in the music app. To delete any songs that you have on there, you can swipe (from right to left) and select Delete (on the actual songs themselves - this doesn't work on albums).
When "Show All Music" is enabled, I don't believe there is any way to remove songs that were purchased through iTunes. If you are not on wifi, you can go to Settings.app -> Cellular -> disable "Music", which will prevent any songs that haven't been downloaded from streaming over cellular data (and which would prevent those songs from playing if that's your desire).
The following is is for iOS 6
Yes, you can.
To delete single songs in Music.app slide from one direction towards the other on a particular song and select "Delete". You can also use this approach with Playlists, Albums, Artists, etc.
If you seeking to delete or edit playlists you can see those buttons at the top of each playlist.
Keep in mind, unless you make the needed changes on your computer, these deleted songs/artists/albums/playlists with sync back to your iPhone.
Actually, there are methods to only check and sync certain songs to your iOS. After connecting your iPhone to iTunes, uncheck the songs you don't want to transfer, then find and check the "Sync only checked songs and videos" box and "Sync only checked songs and videos".
A better solution I personally prefer, is to use something like imElfin iTransfer, which allows me to choose and transfer songs and other stuff very flexibly.
For iOS 11, there is no longer an option under Settings > Music named "Show All Music". Instead, from the Music home, you select "Downloaded Music" row, and the "Songs" and "Playlists" rows under this scene will be filtered accordingly.
I have a similar problem to yours. My smart playlists looked fine on my Mac in iTunes but when I synced them to my iOS device, the smart playlists pulled in songs that I did not want from my iTunes Store purchases. I had unchecked these songs in iTunes on my Mac. On iOS, these unwanted songs had a cloud icon beside their name in my smart playlists. Under Downloaded Music > Playlists, though, my smart playlists appear as I expect.