Shortly after posting my previous notes, SnipURL replied to my problem report:
Snipurl does work in Omniweb. Do you have both Javascript and Flash enabled?
Yes, both are enabled.
I just tried SnipURL in Firefox 3.0.3, and it also does nothing.
It certainly looks like SnipURL conflicts with one of my oodles of OS X enhancements. Rather than spend any more time trying to track this down, I'll just right-click the short URL and do a Copy to Clipboard.