DropDownButton is a button with a list attached. The whole button is one surface: clicking it runs its Command and opens the menu.

<mah:DropDownButton Content="Artists" ItemsSource="{Binding Artists}" />

ItemsSource is the content property, so the items can also be written as the button's child element instead of bound.

It is an ItemsControl, and the list is a ContextMenu

DropDownButton derives from ItemsControl and its list is a ContextMenu attached to the button inside the template. That single fact explains most of how it behaves:

• items become MenuItems, not ListBoxItems
• there is no selection — no SelectedItem, no SelectedIndex, no SelectionChanged
• the button's Content is yours and never changes on its own

The template forwards the usual ItemsControl properties to that menu — DisplayMemberPath, ItemTemplate, ItemTemplateSelector, ItemContainerStyle, ItemContainerStyleSelector, ItemStringFormat, ItemsPanel and GroupStyleSelector — so they work as you would expect even though the menu is not the control you are setting them on. GroupStyle entries are copied across in code, and the menu is given MinWidth="{TemplateBinding ActualWidth}" so it is never narrower than the button.

If you need selection or a button whose label follows the choice, use SplitButton instead.

Reacting to a click

There are two separate things to hook, and they fire in this order:

The button click and the menu are not alternatives. Looking at ButtonClick, the control runs the command, opens the menu if it has any items, and then raises Click:

CommandHelpers.ExecuteCommandSource(this);

if (this.contextMenu?.HasItems == true)
{
    this.SetCurrentValue(IsExpandedProperty, BooleanBoxes.TrueBox);
}

e.RoutedEvent = ClickEvent;
this.RaiseEvent(e);

So a Command on the button runs on every click, including the one that opens the menu. If you only want the item commands, leave Command unset.

DropDownButton implements ICommandSource and overrides IsEnabledCore, so the button disables itself while its command reports CanExecute == false.

Giving the items a command

Each MenuItem gets one entry from the ItemsSource as its DataContext, so a command on the item has to reach back out of that context. An ItemContainerStyle is the usual way, and the way out is through the menu:

<mah:DropDownButton Content="Genres"
                    DisplayMemberPath="Name"
                    ItemsSource="{Binding Genres}">
    <mah:DropDownButton.ItemContainerStyle>
        <Style BasedOn="{StaticResource {x:Type MenuItem}}" TargetType="{x:Type MenuItem}">
            <Setter Property="Command"
                    Value="{Binding RelativeSource={RelativeSource FindAncestor, AncestorType={x:Type ContextMenu}}, Path=PlacementTarget.DataContext.GenreCommand}" />
            <Setter Property="CommandParameter" Value="{Binding Name}" />
        </Style>
    </mah:DropDownButton.ItemContainerStyle>
</mah:DropDownButton>

{Binding Name} reads the genre, because that is the item's DataContext. The other one finds the menu the item sits in and asks what the menu is placed on, which is the button inside the template, and takes the view model from there. Path=DataContext.GenreCommand works as well, since the menu inherits the same view model.

Properties

ButtonStyle and MenuStyle are both registered with Inherits, so setting either one on a parent panel reaches every DropDownButton below it.

Orientation

Vertical stacks the icon above the content and moves the chevron from the right edge to the bottom.

The arrow

<mah:DropDownButton Content="Artists" ArrowVisibility="Collapsed" />

The chevron is a Material ChevronDown path drawn through MahApps.Styles.ContentControl.PathIcon. ArrowMouseOverBrush defaults to the accent brush, which is why the arrow picks up colour on hover while the rest of the button does not.

What a client is told

It is a button that opens and shuts, named after its Content, and a client can open it through the expand pattern rather than having to press it.

Related

SplitButton — the same idea with a selection and a split surface. ContentControlEx is what presents the content, so ControlsHelper.ContentCharacterCasing and ControlsHelper.RecognizesAccessKey apply here too.