I want to take text messages from my iPhone and save them on my computer. Is it possible to save iMessage conversations to my PC?"
Each time you backup your iPhone, text messages are stored by iTunes as part of the iPhone backup, just in case you need to reset your iPhone or restore from backup. These backups aren't searchable, so you can't easily view them on your computer. If you use secure backups, which you should to protect your personal information, the iPhone backup is also encrypted. Thankfully, you can backup your phone for recovery purposes and use a second software app to copy all those SMS text messages from iMessage on your iPhone and transfer them to your computer where you can search and archive the messages however you see fit.