How do I stop purchased music appearing on my iPhone with iOS7?

My Mac has everyone in the families music on it. We have a playlist each that we sync to our various devices.

Since iOS7, all the music I've ever purchased from iTunes is now on my phone, not just my single playlist.

That wouldn't be so bad if the purchased music wasn't dominated by all the rubbish that I've bought for my kids.

I think it's just streaming, but it's still really annoying because my car shuffles between all the songs on the device by default.

Edit: Daryl's suggestion to turn off "Show All Music" in Settings got rid of all the purchased songs I hadn't played. Ones I was foolish enough to play remain firmly stuck on the phone.

Furthermore, these songs cannot be deleted by swiping left to reveal a delete button.

I had the same exact problem. Ended up being that the song had not cpletely finished being downloaded to the phone, so it could be deleted. Go into settings and tell it to display all music again. Then go back to each song and click the cloud icon to download again. After each song downloads you will have the ability to swipe and delete, just like all of your other songs. I'm running ios7 too, so I'm sure this is your problem too.

iPhones can’t stream music directly from iCloud, therefore all music you at some point played on your iPhone has been downloaded to your iPhone.

In order to delete a track, you can go into the song view and swipe the song to the left – a Delete button should appear, allowing you to delete the track from your phone.

One tip that has not been shown in answers thus far is to restart the music app after making the changes to settings. It seems that some residual setting is keeping your music visible and not allowing your settings to be used properly. Please check out the link here which pretty much summarises the same steps as answers already provided, but with pics, and shows where and how to restart the Music app. Also note, that the steps to swipe to delete the music need to be made while the show all music option is enabled, then you can disable it after you have removed them.