Same issue here. No response (no quick popups, etc.) and no instance already running. Tried starting the app from both %localappdata%\Sourcetree\SourceTree.exe and %localappdata%\Sourcetree\app-2.3.1\SourceTree.exe to no avail. Here's what showed up in %localappdata%\Atlassian\SourceTree\sourcetree.log:
ERROR [2017-09-13 16:23:43,362] [1] [SourceTree.App] [.ctor] - finish EnsureSquirrelExecutionStubIsCopied
ERROR [2017-09-13 16:23:43,507] [1] [SourceTree.App] [OnStartup] - Failed to start
System.NullReferenceException: Object reference not set to an instance of an object.
at SourceTree.Notifications.NotificationsManager.SetOwner(NotificationDialogWindow notificationWindow)
at SourceTree.Notifications.NotificationsManager.ShowNotificationDialog[T](NotificationDialogWindow notificationWindow, Tuple`2 customAction, VistaTaskDialogIcon icon)
at SourceTree.Notifications.NotificationsManager.ShowNotificationDialog[T](String title, String message, Tuple`2 customAction, String cancelLabel, String suppressionSetting, Action`1 suppressionChangedAction, Object contentControl, String contentCommandLabel, Action contentAction)
at SourceTree.Notifications.NotificationsManager.ShowNotificationDialogWithYesConfirmation(String title, String message, String details)
at SourceTree.Configuration.WpfSpellCheckerPreFlightCheck.Run()
at SourceTree.AppRoot.RunPreFlightChecks()
at SourceTree.AppRoot.OnStartup(StartupEventArgs e)
at SourceTree.App.OnStartup(StartupEventArgs e)