How do I determine the position of the taskbar on the screen?

There is no real way to do this inside Swing natively, however you can find a possible location with the following ...

GraphicsDevice gd = GraphicsEnvironment.getLocalGraphicsEnvironment().getDefaultScreenDevice();
Rectangle bounds = gd.getDefaultConfiguration().getBounds();
Insets insets = Toolkit.getDefaultToolkit().getScreenInsets(gd.getDefaultConfiguration());

Rectangle safeBounds = new Rectangle(bounds);
safeBounds.x += insets.left;
safeBounds.y += insets.top;
safeBounds.width -= (insets.left + insets.right);
safeBounds.height -= (insets.top + insets.bottom);

System.out.println("Bounds = " + bounds);
System.out.println("SafeBounds = " + safeBounds);

Area area = new Area(bounds);
area.subtract(new Area(safeBounds));
System.out.println("Area = " + area.getBounds());

      

        
        
        
      

    

What are the outputs

Bounds = java.awt.Rectangle[x=0,y=0,width=2560,height=1600]
SafeBounds = java.awt.Rectangle[x=0,y=40,width=2560,height=1560]
Area = java.awt.Rectangle[x=0,y=0,width=2560,height=40]

      

        
        
        
      

    

For my system (mind you, my taskbar is at the top of the screen)

Update

As shown in my previous answer to your question about tray icons ...

public class TestTaskIcon {

  public static void main(String[] args) {

    EventQueue.invokeLater(new Runnable() {
      @Override
      public void run() {

        Image img = null;
        try {
          img = ImageIO.read(new File("floppy_disk_red.png"));
        } catch (IOException e) {
          e.printStackTrace();
        }
        TrayIcon ti = new TrayIcon(img, "Tooltip");
        ti.addMouseListener(new MouseAdapter() {
          @Override
          public void mouseClicked(MouseEvent e) {
            Rectangle bounds = getSafeScreenBounds(e.getPoint());
            JPopupMenu popup = new JPopupMenu();
            popup.add(new JLabel("hello"));

            Point point = e.getPoint();

            int x = point.x;
            int y = point.y;
            if (y < bounds.y) {
              y = bounds.y;
            } else if (y > bounds.y + bounds.height) {
              y = bounds.y + bounds.height;
            }
            if (x < bounds.x) {
              x = bounds.x;
            } else if (x > bounds.x + bounds.width) {
              x = bounds.x + bounds.width;
            }

            if (x + popup.getPreferredSize().width > bounds.x + bounds.width) {
              x = (bounds.x + bounds.width) - popup.getPreferredSize().width;
            }
            if (y + popup.getPreferredSize().height > bounds.y + bounds.height) {
              y = (bounds.y + bounds.height) - popup.getPreferredSize().height;
            }
            popup.setLocation(x, y);
            popup.setVisible(true);
          }
        });
        try {
          SystemTray.getSystemTray().add(ti);
        } catch (AWTException ex) {
          Logger.getLogger(TestTaskIcon.class.getName()).log(Level.SEVERE, null, ex);
        }
      }
    });
  }

  public static Rectangle getSafeScreenBounds(Point pos) {

    Rectangle bounds = getScreenBoundsAt(pos);
    Insets insets = getScreenInsetsAt(pos);

    bounds.x += insets.left;
    bounds.y += insets.top;
    bounds.width -= (insets.left + insets.right);
    bounds.height -= (insets.top + insets.bottom);

    return bounds;

  }

  public static Insets getScreenInsetsAt(Point pos) {
    GraphicsDevice gd = getGraphicsDeviceAt(pos);
    Insets insets = null;
    if (gd != null) {
      insets = Toolkit.getDefaultToolkit().getScreenInsets(gd.getDefaultConfiguration());
    }
    return insets;
  }

  public static Rectangle getScreenBoundsAt(Point pos) {
    GraphicsDevice gd = getGraphicsDeviceAt(pos);
    Rectangle bounds = null;
    if (gd != null) {
      bounds = gd.getDefaultConfiguration().getBounds();
    }
    return bounds;
  }

  public static GraphicsDevice getGraphicsDeviceAt(Point pos) {

    GraphicsDevice device = null;

    GraphicsEnvironment ge = GraphicsEnvironment.getLocalGraphicsEnvironment();
    GraphicsDevice lstGDs[] = ge.getScreenDevices();

    ArrayList<GraphicsDevice> lstDevices = new ArrayList<GraphicsDevice>(lstGDs.length);

    for (GraphicsDevice gd : lstGDs) {

      GraphicsConfiguration gc = gd.getDefaultConfiguration();
      Rectangle screenBounds = gc.getBounds();

      if (screenBounds.contains(pos)) {

        lstDevices.add(gd);

      }

    }

    if (lstDevices.size() > 0) {
      device = lstDevices.get(0);
    } else {
      device = ge.getDefaultScreenDevice();
    }

    return device;

  }
}