Prior iOS versions in your iPhone or other Apple Device used to allow you to easily delete individual text messages and you can still do that with iOS 7, however you just have to do it in a different manner. In order to delete individual text messages or a group of them in your iPhone iOS 7, tap and hold the message that you want to delete when you are in the conversation view. When you do this, a pop up message will show up that says, “Copy/More”, tap on the “More” and selection boxes will show up beside the messages along with an option at the bottom that will allow you to delete or forward whatever messages you choose. This way you can delete all the messages, or just particular ones depending on what you want.
Source iLounge