Ajax Function on a search form not working in Safari & IE
Try attaching the listener on the forms submit event instead, and prevent that instead of just the click on the button. Something like this (make sure you change the selector for your form):

{% if not craft.request.isAjax %}
    {% set pagingJs %}
        $(function () {             
            $('#myForm').on('submit', function (e) {
                e.preventDefault();

                $( ".searchResult" ).remove();
                $( ".noResult" ).remove();

                var search = $('#searchField').val();

                $('#loading').show();
                $.get( "/?q="+search, function( data ) {
                    data = $('#searchQuery', data).html();
                    $( "#searchQuery" ).append(data);
                });
            });
        });
        $( document ).ajaxStop(function() {
           $('#loading').hide();
        });
    {% endset %}
{% includeJs pagingJs %}
{% endif %}