I cannot change my default mail application. The new selection is never saved.

Triple-click anywhere in the line below on this page to select it:

~/Library/Preferences/com.apple.LaunchServices/com.apple.launchservices.secure.plist

Right-click or control-click the highlighted line and select

Services ▹ Show Info in Finder (or just Show Info)

from the contextual menu.* An Info dialog should open.

☞ Does the dialog show "You can read and write" in the Sharing & Permissions section?

☞ In the General section, is the box labeled Locked checked?

☞ What is the Modified date?

If you don't have read and write access to the item, change the settings as directed here. Note, however, that if one file has wrong access settings, most likely others do as well. If the item is locked, unlock it.

*If you don't see the contextual menu item, copy the selected text to the Clipboard by pressing the key combination command-C. Open a TextEdit window and paste into it (command-V). Select the line you just pasted and continue as above.

Although the problem is not the same, please follow the instructions in this support article to reset the Launch Services database. Instead of typing the shell command directed in the article, I suggest that you copy and paste it from the web page into the Terminal window. Make sure you select the whole line. One way to do that is by triple-clicking anywhere in the line.

If you don't already have a current backup, back up all data before proceeding. You may have to log out or restart to see a change.