Aligning controls on both left and right side in a stack panel in WPF

Just do not use a StackPanel, StackPanels stack. They do, for obvious reasons, not allow alignment in the direction in which they stack. Use a Grid, with column definitions like so:

<Grid.ColumnDefinitions>
    <ColumnDefinition Width="Auto" />
    <ColumnDefinition Width="*" />
    <ColumnDefinition Width="Auto" />
</Grid.ColumnDefinitions>

Even though this is old, should someone come across this like I did, here's a simple solution.

Make a new grid and inside that grid put two stack panels with different Horizontal Alignment.

<Grid>
    <StackPanel Orientation="Horizontal" HorizontalAlignment="Left">
        <!--Code here-->
    </StackPanel>
    <StackPanel Orientation="Horizontal" HorizontalAlignment="Right">
        <!--Code here-->
    </StackPanel>
</Grid>

The possible issue is that now without extra handling the two could overlap with each other.

As you have set the StackPanel's orientation to Horizontal, the HorizontalAlignment property won't work on child-elements. You can keep the StackPanel if you need additional controls, though I would recommend switching to a Grid (among other things) to build the layout you want.

Also, the Grid will allow you to control the actual width of each column:

<Grid>
    <Grid.ColumnDefinitions>
        <ColumnDefinition Width="50" />
        <ColumnDefinition Width="150" />
    </Grid.ColumnDefinitions>

    <RadioButton
        Grid.Column="0"
        ...
    />

    <Label
        Grid.Column="1"
        ...
    />
</Grid>