There's no point doing a comprehensive search, because we don't have the storage capacity for all those primes. At least until Google makes a free online service with infinite gigabytes, then we'll continue.
Anything we can guess about the distribution of primes, we can guess from the ones already known. If someone wants a certain sized prime, there are free programs for that. Primes of special forms are more interesting, because we can search for them faster, and they might be good for something sometime (a good excuse for anything in mathematics).