I have the EXACT same problem, and it also started TODAY.
I had no problems downloading apps on my iphone through the appstore a few days ago.
The last app i downloaded was mirrors edge a couple of days ago when it was free.
I have a first gen iphone with OS 3.1.3 ( last available OS for my phone).
I have tried powering it off and back on several times.
I have logged in and out of my apple account within settings, and within the appstore itself
A few extra notes:
App updates through the appstore on my phone still works, i can click on update all and it then updates them.
I updated a couple of apps this morning ( Battery Doctor,Gamefly ).
As of a few minutes ago there is now a new app (Airmiles) waiting to be updated, but i held off on updating it.
So it looks like updating apps through the appstore on my iphone still works, but i cannot download any other new apps.
Conncting to my PC and itunes i can sync my iphone and it pushes apps to my PC.
I can download apps on my PC and push them to my iphone.
I have plenty of free space and icon positions/pages for more apps on my phone.
Its just the appstore on my iphone that NO-Longer works!
The only thing i have not tried yet ( and i dont want to do ) is a factory reset.
This is starting to look like some kind of appstore problem.
I would really like a solution to this problem.
PS. im from Canada, ( in case its only a Canada Appstore problem )