Magento ver. 1.9.3.1 Continue Button not moving to next section (Payment) during chec

We have shopping website developed using Magento ver. 1.9.3.1. We are facing problem in checkout process. Page is accepting billing information correctly and after pressing ‘Continue’ button, the control moves to next page that is ‘Shipping’ page (See image Step 1.png). On selection of available shipping rates and pressing ‘Continue’ button, the control does not move ahead (See Image Step 2.png). It’s kind of stuck here.

We need developer to fix this error and make sure it works well on chrome, IE as well on firefox. Developer also make sure that it is working fine for ‘Guest’ account as well as ‘Logged-in’ User Account (See Other1.png)