How to set options of a dropdown using Javascript/jQuery?

You don't even necessarily need jQuery:

var select = document.getElementById("D1"),
    opt = document.createElement("option");
opt.value = "value";
opt.textContent = "text to be displayed";
select.appendChild(opt);

Example.

But here it is with jQuery anyway:

$("select#D1").append( $("<option>")
    .val("value")
    .html("text to be displayed")
);

Example.

There are a few different ways. One is:

$("#D1").append("<option>Fred</option>");

Here's yet another way to remove options using javascript (similar to Andrew Whitaker's answer) by using the options array:

var select = document.getElementById("D1");
select.options.length = 0;//remove all options

Yet another way of doing it, using select's add method:

var select = $("#select")[0];

select.add(new Option("one", 1));
select.add(new Option("two", 2));
select.add(new Option("three", 3));

Example: http://jsfiddle.net/pc9Dz/

Or another way, by directly assigning values to the select's options collection:

var select = $("#select")[0];

select.options[0] = new Option("one", 1);
select.options[1] = new Option("two", 2);