The fourth of your attempts is the easiest way to get the selected value of the select element, and it should work.
var inventory_rule = $("#inventory_rule").val();
I wrote a quick and dirty example for you on jsfiddle.net, showing that it works. This means that your selector is probably wrong. Check and verify that the select element has id="inventory_rule" and make sure that the id attribute is also unique on the page . Do not forget the var keyword if this is the first time you declare a variable.
EDIT: highlight the part that the id attribute is unique, not a unique identifier will certainly cause problems in IE.
source share