I've struggled for weeks with the 'spinning circle' at checkout with all browsers.
I tried all the usual 'fixes' mentioned about IE9 compatibility, uninstall conflicting plugins, jQuery probs, disable https, make the url the same in both boxes of WP settings, increase memory limit, check it when logged in/out...NONE OF THESE WORKED (for me).
As standard I install WordPress and then follow the hardening guide. For me this was the problem, WooCommerce requires access to /wp-admin/ at checkout for some of the AJAX code to work for the checkout form/calcs at the bottom.
The fix: Make sure you do not have any blocks to /wp-admin/ in .htaccess, httpd.conf files or in the web server config anywhere. Allow all public access to /wp-admin/ and then retest, this immediately fixed it for me.
Hope this helps some people
Neil [sig moderated as per the Forum Rules]