ListBox already contains ScrollViewer. By default the ScrollBar will show up when there is more content than space. But some containers resize themselves to accommodate their contents (e.g. StackPanel), so there is never "more content than space". In such cases, the ListBox is always given as much space as is needed for the content.
In order to calculate the condition of having more content than space, the size should be known. Make sure your ListBox has a constrained size, either by setting the size explicitly on the ListBox element itself, or from the host panel.
In case the host panel is vertical StackPanel and you want VerticalScrollBar you must set the Height on ListBox itself. For other types of containers, e.g. Grid, the ListBox can be constrained by the container. For example, you can change your original code to look like this:
<Grid Name="grid1">
<Grid>
<Grid.RowDefinitions>
<RowDefinition Height="2*"></RowDefinition>
<RowDefinition Height="*"></RowDefinition>
</Grid.RowDefinitions>
<ListBox Grid.Row="0" Name="lstFonts" Margin="3"
ItemsSource="{x:Static Fonts.SystemFontFamilies}"/>
</Grid>
</Grid>
Note that it is not just the immediate container that is important. In your example, the immediate container is a Grid, but because that Grid is contained by a StackPanel, the outer StackPanel is expanded to accommodate its immediate child Grid, such that that child can expand to accommodate its child (the ListBox).
If you constrain the height at any point — by setting the height of the ListBox, by setting the height of the inner Grid, or simply by making the outer container a Grid — then a vertical scroll bar will appear automatically any time there are too many list items to fit in the control.