So recently, SSMS started exiting on startup for me. It would display the splash screen, and then about the time the connection dialog was to show, it would just exit. Initially I tried repair, then re-installing, erasing its section in the registry, and the files in AppData/Local, nothing worked.

I checked the Windows Event log and found the following exception:

Application: Ssms.exe
Framework Version: v4.0.30319
Description: The process was terminated due to an unhandled exception.
Exception Info: System.ArgumentNullException
   at System.Windows.Forms.ComboBox+ObjectCollection.AddInternal(System.Object)
   at System.Windows.Forms.ComboBox+ObjectCollection.Add(System.Object)
   at Microsoft.SqlServer.Management.UI.ConnectionDlg.LoginControl.UpdateServerInstanceMruOnly()
   at Microsoft.SqlServer.Management.UI.ConnectionDlg.LoginControl.UpdateServerInstanceMruAndValue()
   at Microsoft.SqlServer.Management.UI.ConnectionDlg.LoginControl.OnSTIndexChange(System.Object, System.EventArgs)
   at System.Windows.Forms.ComboBox.OnSelectedIndexChanged(System.EventArgs)
   at System.Windows.Forms.ComboBox.set_SelectedIndex(Int32)
   at System.Windows.Forms.ComboBox.set_SelectedItem(System.Object)
   at Microsoft.SqlServer.Management.UI.ConnectionDlg.LoginControl.InitFromLastUsedServerType()
   at Microsoft.SqlServer.Management.UI.ConnectionDlg.ConnectionDialog.DoOneTimeInitialization()
   at Microsoft.SqlServer.Management.UI.ConnectionDlg.ConnectionDialog.OnVisibleChanged(System.EventArgs)
   at Microsoft.SqlServer.Management.UI.VSIntegration.ShellConnectionDialog.OnVisibleChanged(System.EventArgs)
   at System.Windows.Forms.Control.WmShowWindow(System.Windows.Forms.Message ByRef)
   at System.Windows.Forms.Control.WndProc(System.Windows.Forms.Message ByRef)
   at System.Windows.Forms.ScrollableControl.WndProc(System.Windows.Forms.Message ByRef)
   at System.Windows.Forms.Form.WmShowWindow(System.Windows.Forms.Message ByRef)
   at System.Windows.Forms.Form.WndProc(System.Windows.Forms.Message ByRef)
   at System.Windows.Forms.Control+ControlNativeWindow.OnMessage(System.Windows.Forms.Message ByRef)
   at System.Windows.Forms.Control+ControlNativeWindow.WndProc(System.Windows.Forms.Message ByRef)
   at System.Windows.Forms.NativeWindow.DebuggableCallback(IntPtr, Int32, IntPtr, IntPtr)

This made me think, the error appears to be on loading a combo box, perhaps the combo that has the connections. So where are the list of connections stored? After some googling I found this stack overflow answer Where does SSMS store its server names

So for 18.x, this is in C:\Users\[USERNAME]\AppData\Roaming\Microsoft\SQL Server Management Studio\18.0\UserSettings.xml

I renamed the file to UserSettings.xml.old, started SSMS and voila!, it created a new user settings file and I was good to go

Running both XML files through Beyond Compare, I found this section which I believe to be the most likely the culprit: