Formkey validation is not enabled. I was on 188.8.131.52 and everything was running perfectly. Now, after the upgrade, checkout just won't move to Step 3. I hit continue, it says loading next step and doesn't go to the next step no matter how many times you hit continue.
There were several reports of that kind. I'm not sure about one final soolution becuase I've saw different problems/solutions several times.
Maybe this links can help you to start:
May be you have enabled the form key validation in the configuration for the checkout and you are still using the billing.phtml and shipping.phtml in your frontend design under template\checkout\onepage where the following line is missing <?php echo $this->getBlockHtml('formkey') ?>
There are multiple reason for this, you can debug error like this.
1. Confirm that you are proper implementing the form key validation into all the places into checkout.
2. You can check the consol, does any error show there? If yes then try to resolve that error.