<p>I have a select element in a form, and I want to display something only if the dropdown is not visible.  Things I have tried:</p>

<ul>
<li>Watching for click events, where odd clicks mean the dropdown is visible and even clicks mean the dropdown isn't.  Misses other ways the dropdown could disappear (pressing escape, tabbing to another window), and I think this could be hard to get right cross-browser.</li>
<li>Change events, but these only are triggered when the select box's value changes.</li>
</ul>

<p>Ideas?</p>

I don't think there's direct support.  You could also sit on the onblur of the select -- it gets called when the select loses focus.
Depending on how important it is, you could try implementing your own control or starting from one like a drop-down menu control which is similar.  Usually, unless it's critical to your application, it's not worth doing this.  If you decide to go this route, here's a discussion of someone trying to do it using dojo as a basis:
http://dojotoolkit.org/forum/dijit-dijit-0-9/dijit-support/emulating-html-select