How to set default web browser per domain?

My default web browser is Opera. This is the desired behavior for general browsing.

However, I would prefer that a URL in a domain opens in Internet Explorer where the content of that domain is more compatible with Internet Explorer than with Opera.

For example, this link:

http://ssrs/Reports

should open in Internet Explorer because the location is in the ssrs domain. In my organization, content in this domain is served by SQL Server Reporting Services, which works well only with Internet Explorer.

This link:

http://fogbugz/default.asp

should open in Internet Explorer because the location is in the fogbugz domain. In my organization, content in this domain is served by FogBugz. Several features of FogBugz are broken or buggy in Opera.

Any other domain should open in Opera.

How do I configure Windows 7 with these rules?

Browser Chooser 2 should do everything you need. From the site:

Browser Chooser 2 is a small tool acting as the default browser allowing you to choose what browser or other tool to open any given link in.

However, it can do a lot more. By default it gives a pop-up similar to TurnOut.NET but is far more configurable. It will let you add URLs with wildcards and regular expression matching, and automatically open it in the corresponding browser (see screenshot).

It's a beta release and a little unpolished, but it should fit the bill.

There can't be a simple system-wide solution. Here is why:

• Windows will always recognize your http://ssrs/Reports link as HTTP-protocol and open it with the registered application regardless of the rest of the link.
• you can associate another browser which windows 7 should use for HTTP-URLs.
  But that won't be conditional since you can only choose one at a time.
• you can modify this association via regedit.exe to point it to a (portable) software

Your best shot will be to write (or ask a developer for) a short AutoHotKey tool and point your HTTP-protocol to that software via regedit. The link shows the basic steps.
After that your new tool will get every URL as parameter and can choose what to do next

if parameter like http://ssrs/* then 
   open "C:\Program Files\Internet Explorer\iexplore.exe"
else
   open "C:\Program Files\Opera\opera.exe"
end

My AHK days are long gone.
But maybe another superuser can help you out.

For Mac: Choose Wisely

For Windows: TurnOut.NET

Assuming you are in a corporate environment and are willing to use Google Chrome, there is a feature called Legacy Browser Support (LBS).
This uses group policy to force certain URLs to open in an alternate browser (such as IE). This is the LBS extension add-on store home page

There is to much detail to add it all here so here are the linked pages:
Step 1 Read this source information page Legacy Browser Support.
Step 2 Then install from 'Install LBS Chrome extension'
Step 3 Configure which browsers to open with LBS
Step 4 feedback and roll out

This should work regardless of if the user types in the URL or clicks on a link in an email application: