The site looks good and works well. However, are ~4000 nonmember domains and ~13,000 member viewable domains "all" one can get? That is, Domain Pigeon pre-selected these names? If so, that is certainly a different approach, going for quality instead of quantity(since 13,000 domain names is certainly a drop in an ocean of possible domain names not taken yet).
I kept looking for some sort of associative search box, for example one could enter "LISP" and find "LISPy" names.
When domains are added they are randomly assigned to be either community domain names, which everyone can see, or private, which only members can see. Currently its about 3 private for everyone 1 community. This is going to change at some point though in favor of some better distribution system (imagine the case where Domain Pigeon has 250K public, 750K private--see the problem?).
As far as searching goes, the box on the left column searches the database and will display domains that match your search term. Since the site is relatively new and I've focused mostly on abstract domain names, most won't find any results (which is also why the search feature currently isn't in a more prominent position.)
There are going to be lots of positive changes in the coming weeks, including what I hope will be the best domain name generator on the net. Stay tuned...
Why do you keep the good domains private? I bet the affiliate commissions from showing off these domains would be much higher than the revenue from membership fees.