JimiE, you can find an abundance of "recommendations" . . cough . . cough . . promotional posts on this topic at DigitalPoint.
If you want to see lists of directories you might research "directory submission" services.
Also, you might search DMOZ for directories in whatever verticals you are pursuing.
A quick few Google searches also created some leads: network of directories, directory network, etc.
Of course there have been many discussions about the merits or lack thereof of indiscriminate directory listings . . IFF pan-networks submission is your goal.
Food for thought: Many so-called "apps" are actually some form of database driven listing / directory service. I'd expand my thinking to see which apps are popular.
Finally, if you are looking for ideas about what .directory domains to register . . eh . . meh . . good luck with that idea. :p, ;)
Having pointed you to DP and provided the best alternative ways to address your quesion I will now lock this thread to avoid people posting links to their directories or networks. Sorry, but it's the way we roll. We're open to discussing all manner of directory related issues (see this forum's Charter) but we're rarely open for the business of promoting individual directories.