Apparently popups don't currently support drop shadows, see link.
However, I have come up with a workaround this which works rather well IMO. Basically the idea is to have a Canvas nested within another transparent Canvas and just apply the drop shadow to the nested Canvas. Simple. Heres an example:
<Grid>
<TextBox x:Name="MyTxtBx" Width="50"
Height="20" Text="Hello"/>
<Popup IsOpen="True" Width="200" Height="100"
PlacementTarget="{Binding ElementName=MyTxtBx}"
AllowsTransparency="True" >
<Canvas Background="Transparent">
<Canvas Background="Green" Width="150" Height="50">
<Canvas.BitmapEffect>
<DropShadowBitmapEffect Softness=".5"
ShadowDepth="5"
Color="Black"/>
</Canvas.BitmapEffect>
<Label Content="THIS IS A POPUP TEST"/>
</Canvas>
</Canvas>
</Popup>
</Grid>
The points to note are that the nested canvas needs to be smaller than the size of it's container. Also AllowsTransparency must be set too.