PrimeGrid once ran the primegen project which actually did a comprehensive sequential search of all numbers from 3 on up. You can torrent the final results here. The page indicates that the upper bound for the numbers is 210 000 000 000. However, thats minuscule compared to the numbers searched by Cullen-Woodall/TPS/etc., because primes of certain specific forms can be checked much more efficiently.

The primegen project was retired though and I am unaware of anyone else who is continuing to do a sequential search and/or searched a larger space than PrimeGrid.
____________

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).