When recently working with a customer we ran across problems logging on to SharePoint Online sites.  When accessing a SharePoint site collection the user reported the screen "flickered" and then it displayed an error message.  Upon further investigation we found the "flickering" was actually the browser bouncing back and forth between URL's in the address bar.  The address bar also changed its background color from white to green which gave it a flicker or flash effect. The error displayed was:

Microsoft Online Services is unavailable from this site
for one of the following reasons:

• This site may be experiencing a problem
• The site may not be a member of the Windows
  Live Network

You can:

• You can sign in or sign up at the other Microsoft
  Online Services site, or try again later at this site

Here's a screenshot of the error:

My translation of the error message is this: the security settings need to be changed in your browser…

The solution to problems logging on to SharePoint Online sites is to add https://*.microsoftonline.com and https://*.sharepoint.com to Trusted Sites in IE.

Here are the steps to accomplish this:

1. In Internet Explorer, on the Tools menu, click Internet options.
2. Click the Security tab, click the Trusted sites zone, and then click Sites.
3. Verify that the following entries are listed in the Websites list:
   https://*.sharepoint.com
   https://*.microsoftonline.com
4. If these entries are not present, add them to the Trusted sites list of websites.
5. After you've added the sites, click Close and then Ok in the Internet Options window.
6. Close and re-open the browser and try logging on again.

With these sites added to the Trusted Sites zone the appropriate security settings are configured to allow for a user to login to the SharePoint Online sites.