How to Backup and Restore Signatures in Outlook 2013
Please note that the signatures used by Outlook are not stored in the .pst file Outlook uses for email messages. They are stored at: C:\Users\%username%\AppData\Roaming\Microsoft\Signatures

To open this location from within Outlook, start by clicking on the File tab.

On the Account Information screen, click on Options in the left-hand column

Click on Mail

In the Compose messages section, hold down CTRL and left click the Signatures button.

The Signatures folder opens up in Windows Explorer. To backup the files, just copy them to another location (i.e. a network share or somewhere where you can access them later).
To restore your signatures, copy them from your backup location back into the C:\Users\%username%\AppData\Roaming\Microsoft\Signatures location. The next time you open Outlook, your saved signatures will be available. Please note that if you had signatures selected as default signatures for new emails and for replies and forwards, you'll need to define those again.
IMPORTANT NOTE: Each signature you create in Outlook exists in 3 formats: Plain Text (.txt), HTML (.htm), and Rich Text (.rtf). You'll need to backup all these formats for each signature.