<p>Is there a way in Delphi 7 to find out if a pop-up menu is visible (shown on the screen) or not, since it lacks a Visible property.</p>

You could make your own flag by setting it in the OnPopup event. The problem is knowing when the popupmenu is closed.
Peter Below
has a solution for that.
But my I ask why you would want this? Maybe there is a better way to solve the underlying problem.