Disable selecting in WPF DataGrid

Simply add IsHitTestVisible="False" to DataGrid definition.

The clean way would be, to just override the styles of the row and the cell

<DataGrid.Resources>
    <ResourceDictionary>
        <Style x:Key="{x:Type DataGridCell}" TargetType="{x:Type DataGridCell}">
            <Setter Property="Background" Value="{x:Null}" />
            <Setter Property="BorderBrush" Value="{x:Null}" />
            <Style.Triggers>
                <Trigger Property="IsSelected" Value="True">
                    <Setter Property="Background" Value="{x:Null}" />
                    <Setter Property="BorderBrush" Value="{x:Null}" />
                </Trigger>
            </Style.Triggers>
        </Style>
        <Style TargetType="{x:Type DataGridRow}">
            <Setter Property="Background" Value="{x:Null}" />
            <Setter Property="BorderBrush" Value="{x:Null}" />
            <Style.Triggers>
                <Trigger Property="IsSelected" Value="True">
                    <Setter Property="Background" Value="{x:Null}" />
                    <Setter Property="BorderBrush" Value="{x:Null}" />
                </Trigger>
            </Style.Triggers>
        </Style>
    </ResourceDictionary>
</DataGrid.Resources>

To completely disable selection of rows in a DataGrid, you could do the following:

<DataGrid>
    <DataGrid.RowStyle>
        <Style TargetType="DataGridRow">
            <Setter Property="IsHitTestVisible" Value="False"/>
        </Style>
    </DataGrid.RowStyle>
    <!--Other DataGrid items-->
</DataGrid>

This could be considered more favorable than setting <Setter Property="IsEnabled" Value="False"/> due to the fact that doing the aforementioned technique causes the style of the row to change. It also does not disable context menus from appearing when right-clicking.

Lastly: it is important to note that setting "IsHitTestVisible" to "False" disables all interaction with the rows, including editing.

However, if all you want to do is change the styling of the row when selected, please view the answers here.