Been pulling my hair out with this one for over an hour.
I have exactly the same issue with my latest WordPress installation (every other one to date works fine).
Clicking on Media then Add New offers a button 'Select Files' (which does nothing when clicked on) and the following text:
You are using the multi-file uploader. Problems? Try the browser uploader instead.
Maximum upload file size: 2MB. After a file has been uploaded, you can add titles and descriptions.
In the above, a text link is offered on 'browser uploader' but clicking on it does nothing. Except that is to append a hash # symbol to the url (wp-admin/media-new.php becomes wp-admin/media-new.php#) at the top of the screen.
I've tried everything, uninstalled both plugins (All in One SEO Pack and Custom Permalinks), reverted to default Twenty Eleven theme, created a directory called uploads and ensured that its permissions are writeable.
This is a website with no content yet! I've set up a home page but added nothing. It's barely a couple of hours old and I downloaded the latest version WordPress 3.4.2 on Monday this week. There is no customisation.
Why on this website and none of my others? I've tried to get the Media Uploader to work using Safari, Chrome, Firefox and Internet Explorer. Same thing - it just doesn't work. If anyone can help me I would be very grateful indeed.