From comments of OP, this explains how to debug the checkout steps.
Browse to your checkout.
Hit 'f12' which will open the chrome developer tools.
At the top, click the 'console' icon.
Perform your action(s) and see if any error sre reported in the console. If so, deal with them. they will be js errors, so nothing strange there.
To see what the responses are on actions:
Hit the 'network' icon in the developers tools.
Then at the bottom of that panel, click ' 'XHR' filter option. This will only then show you the ajax requests sent, and filter out the noise.
Do your action, and you will see the ajax call request logged in the netwrok tab
Now you can click on any of the requests to view detailed info.
Click on the 'response' tab to see the html returned.
If your ajax actions caused any php errors, they will (should) be there, or an error response if the ajax call handled them correctly

if this does not lead to your solution, you will need to use the 'sources' tab to do some breakpoints and debug the checkout javascript.
Will stop there, as you may not need go that far. :)
EDIT: Debug on live site on OP request (I had some time to kill to help a fellow South African out ;) )
You seem to have created customization to the OPC payment methods file located here in a base magento install: design/frontend/base/default/template/checkout/onepage/payment/methods.phtml
The file starts with this code (normally)
<dl class="sp-methods" id="checkout-payment-method-load">
and that is missing from your checkout OPC display rendering.
In the opcheckout.js file the following line of code tries to update the checkout payment methods, via the ajax result gotten.
if (response.update_section) {
$('checkout-'+response.update_section.name+'-load').update(response.update_section.html);
}
response.update_section.name is populated with 'payment-method'
Thus the code is trying to update the element with the id of
checkout-payment-method-load
but since it does not exist, bails.
Your html display:

what it should look like

Best to get your developer to look what was updated in your site since this broke.