I stumbled across an easy way to find the path(s).
Open the Application Properties.
Under the "Application" tab, select "Assembly Information...".
Change the value of "Company". Select OK to save.
Select the "Settings" application properties tab.
Select "Synchronize" (first button in the options at the top of the tab).
You should then receive a Visual Studio information dialogue saying "No user.config files were found in any of the following locations:" followed by a list of locations where the settings are saved.