In iTunes (12.3 on Windows), clicking
Edit > Preferences... > Device tab
shows an empty list of backups even though there should be a bunch of them.
I needed to get my phone replaced, so I backed up my phone last night, took it to Apple today and got a replacement. In trying to restore the new phone from the backup, iTunes never offered the option - it would just say "Welcome to your new iPhone" with a Continue button.
No matter what I did, it would never offer the option to restore from a previous backup. So I selected Continue and it set it up as a new phone. The options to restore from a previous Backup were greyed out. I tried various suggestions from the internet with no luck - for example, doing a backup and then holding down shift when clicking the Restore button without any luck.