Java (6u20) Applet on Windows (XP SP 2|XP SP 3|Vista|7) in IE (7|8) or FF (3.6.3).
If not mentioned explicitly, configurations not mentioned here don't indicate that there is not a problem with those configs. If you want to check whether the bug exists also in your environment, download the testcase and run it.

The problem is that flickering (and sound) occurs whenever software opens two JOptionPanes in a row.

    JOptionPane.showMessageDialog(AppletTest.this, "Test 1");
    JOptionPane.showMessageDialog(AppletTest.this, "Test 2");

The first is ok, but if a second follow-up JOptionPane is displayed it always flickers a short time. If the environment supports sound, the user also hears the warning sound of Windows. That behavior (flickering and warning sound) irritates customers actually.

It is not reproducible when using an Application instead of an Applet.
(StandaloneTest.java is attached)

It is not reproducible on Solaris.

If the second JOptionPane is shown within SwingUtilities.invokeLater is does not (always) flicker. However, that workaround does not always work. And the customer do not wish to use invokeLater every time a JOptionPane should be shown.

    SwingUtilities.invokeLater(new Runnable() {
        public void run() {
            JOptionPane.showMessageDialog(AppletTest.this, "Test 3");
        }
    }

Runnable Testcase is attached. For IE browsers, open AppletTestForIE.html, for FF browsers, open AppletTestForFF.html.