Steven's answer works to disable the page - however, I found that if I used that method, the other elements on the page remained blocked and could not be clicked on. You could get around it with hiding and showing the element as desired - however, when I did that I found that the SpinJS spinner stopped centring and ended up in the top-left corner.
In the end, I combined SpinJS with the JQuery BlockUI Plugin, which puts an overlay over the page temporarily and prevents any clicking, with some customisable options.
So in the end, my code looks something like...
var target = document.getElementById("centre");
var spinner = new Spinner(opts).spin(target);
$.blockUI({ message: null, overlayCSS: { backgroundColor: '#ddd' } });
//Things things things
spinner.stop();
$.unblockUI();