How to capture a mouse click on an Item in a ListBox in WPF?

I believe that your MouseLeftButtonDown handler is not called because the ListBox uses this event internally to fire its SelectionChanged event (with the thought being that in the vast majority of cases, SelectionChanged is all you need). That said, you have a couple of options.

First, you could subscribe to the PreviewLeftButtonDown event instead. Most routed events have a routing strategy of Bubbling, which means that the control that generated the event gets it first, and if not handled, the event works its way up the visual tree giving each control a chance at handling the event. The Preview events, on the other hand, are Tunneling. This means that they start at the root of the visual tree (generally Window), and work their way down to the control that generated the event. Since your code would get the chance to handle the event prior to the ListBoxItem, this will get fired (and not be handled) so your event handler will be called. You can implement this option by replacing MouseDoubleClickEvent in your sample with PreviewMouseLeftButtonDown.

The other option is to register a class handler that will be notified whenever a ListBoxItem fires the MouseLeftButtonDown event. That is done like this:

EventManager.RegisterClassHandler(typeof(ListBoxItem),
    ListBoxItem.MouseLeftButtonDownEvent,
    new RoutedEventHandler(this.MouseLeftButtonDownClassHandler));

private void OnMouseLeftButtonDown(object sender, RoutedEventArgs e)
{
}

Class Handlers are called before any other event handlers, but they're called for all controls of the specified type in your entire application. So if you have two ListBoxes, then whenever any ListBoxItem is clicked in either of them, this event handler will be called.

As for your second question, the best way to know what type of event handler you need for a given event, and to see the list of events available to a given control, is to use the MSDN documentation. For example, the list of all events handled by ListBoxItem is at http://msdn.microsoft.com/en-us/library/system.windows.controls.listboxitem_events.aspx. If you click on the link for an event, it includes the type of the event handler for that event.

There is also another way - to handle PreviewMouseDown event and check if it was triggered by the list item:

In XAML:

<ListBox PreviewMouseDown="PlaceholdersListBox_OnPreviewMouseDown"/> 

In codebehind:

private void PlaceholdersListBox_OnPreviewMouseDown(object sender, MouseButtonEventArgs e)
{
    var item = ItemsControl.ContainerFromElement(sender as ListBox, e.OriginalSource as DependencyObject) as ListBoxItem;
    if (item != null)
    {
        // ListBox item clicked - do some cool things here
    }
}

Was inspired by this answer, but it uses listbox by name, I propose to use sender argument to avoid unnecessary dependencies.