Another vote for vertical.
Build a rack - don't just lean boards against the wall. With several cross braces, you won't ever have an issue with bowing.
I prefer to stack the boards on edge (like leaves in a book). This makes looking through the boards easier.
A few supports can also be place lower down behind the rack for some horizontal storage.
I also use horizontal storage, but only for long term boards I know I won't be using anytime soon.
Another option is you have room is build an A frame.