How can I detect Windows 10 light/dark mode?

Before the Windows Anniversary update you could not do that. The application theme was always the one you set in the App.xaml file:

<Application
  ...
  RequestedTheme="Dark">
</Application>

Now with the new Anniversary Update, you can remove this line from the App.xaml file, which will make the app honor the user's system settings.

The RequestedTheme enumeration has actually three values - Dark, Light and Default. Default is the value that reflects the system settings, Dark and Light force the theme.

If you want to actually detect the current theme in code when App's RequestedTheme is Default, you will probably need to check the some color resource like SystemAltHighColor for its value, because that will give you an idea of what theme is currently set.

For Windows 10, the value of the AppsUseLightTheme property in the path HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Themes\Personalize of the registry specifies wherever Windows is in dark or light mode.

I have found an easier solution, which should work in JavaScript apps as well, without requiring the Windows Runtime Component - the UISettings class:

var uiSettings = new Windows.UI.ViewManagement.UISettings();
var color = uiSettings.getColorValue(
                        Windows.UI.ViewManagement.UIColorType.background
                       );

The color you get is either black for dark theme or white for light theme.

The class also has very useful event ColorValuesChanged which you can use to observe theme changes at runtime.

You can create a Windows Runtime Component project in your solution from there you access Windows.UI.Xaml namespace. Add a method to check current ApplicationTheme like that.

public sealed class Test
{
    public static string CurrentTheme()
    {
        var isDark = Application.Current.RequestedTheme == ApplicationTheme.Dark;

        if (isDark)
            return "Dark";

        return "Light";
    }
}

Add reference to windows runtime component project in your javascript app project and you can call this method where ever you want to check application theme. Take a look here for walkthrough on createing Windows Runtime Component.