I just checked the above .htaccess file in a 301 Redirect Checker to see if it's really doing it. It is doing it for the following:
www.mysite.com/default.aspx
www.mysite.com/Default.aspx
redirects both to www.mysite.com/

However it's not doing it for the following (non-www versions):
mysite.com/default.aspx
mysite.com/Default.aspx
Those two redirects to www.mysite.com/default.aspx even though it shows it's redirecting to www.mysite.com/ in the browser address bar.

So I made some changes and it started working for non-www versions as well. Here's the new .htaccess code.

RewriteEngine on
RewriteCond %{HTTPS} (on)?
RewriteCond %{HTTP:Host} ^(?!www\.)(.+)(default\.aspx)?$ [NC]
RewriteCond %{REQUEST_URI} (.+)
RewriteRule .? http(?%1s)://www.%2%3/ [R=301,L]
RewriteRule ^default\.aspx / [NC,R=301,L]

The only change in the second version is the second RewriteCond... I added (default\.aspx)? at the end, which takes care of the non-www version (mysite.com/default.aspx or mysite.com/Default.aspx)

Any opinions if I'm doing it right or wrong? As far the 301 redirect checking tool shows, it is correct but I'm always suspicious of these kinds of tools :)

You are right. Thank you for the tip.

The first version is more sound because %{HTTP:Host} includes Host header value. There should not be any default.aspx. But it works because (default\.aspx)? is not mandatory part. I mean ^(?!www\.)(.+)(default\.aspx)?$ will be matched with mysite.com.

So after my tests, I think the original version I presented is more sound and more practical, which is:

RewriteEngine on
RewriteCond %{HTTPS} (on)?
RewriteCond %{HTTP:Host} ^(?!www\.)(.+)$ [NC]
RewriteCond %{REQUEST_URI} (.+)
RewriteRule .? http(?%1s)://www.%2%3 [R=301,L]
RewriteRule ^default\.aspx$ / [NC,R=301,L]

I just added a $ sign at the end of the last rule.

It might have done it before because of (default\.aspx)? in the end of the HTTP:Host condition. That's why I presented the above code without it to be more sound. It works without a problem.

I'm not sure what might be causing Search to fail with the redirect in your set up because no error is reporting on my site. Search is giving a 200.

It should not give you a 301 redirect unless the button call in search includes default.aspx in the path because that rule(^default\.aspx$) simply tells the server to redirect all requests including default.aspx to the root. Not sure exactly what might be triggering the error on your site.

Did you check the form action of the page? My guess is that the form action is default.aspx, so when you post back, it's requesting that URL but the redirect rules are interfering.

OK, so now Omer's 'magic' works perfectly :) Even the search functionality is working with the redirect. Thanks to everyone for piecing together the puzzle.

RewriteEngine on
RewriteCond %{HTTPS} (on)?
RewriteCond %{HTTP:Host} ^(?!www\.)(.+)$ [NC]
RewriteCond %{REQUEST_URI} (.+)
RewriteRule .? http(?%1s)://www.%2%3 [R=301,L]
RewriteCond %{REQUEST_METHOD} GET|HEAD
RewriteRule ^default\.aspx$ / [NC,R=301,L]