I did spend quite a time figuring out jQuery verbs to do just what I needed. So I’ll share this with my blog and possible readers. On my view I had a model written out, some other stuff, but here I’m only interested in DropDown and JavaScript bits:

@Html.DropDownListFor(m => model.tagId, new SelectList(ViewBag.Tags, "Value", "Text", model.tagId), new {@class = "document-category", data_id = model.modelId})

where

ViewBag.Tags was populated in controller like this:

ViewBag.Tags = _tagRepository.All
    .OrderBy(t => t.Name)
    .Select(t => new { Text = t.Name, Value = t.TagId.ToString() })
    .OrderBy(t => t.Text)
    .ToList();

Back to the view. Added JavaScript like this one:

$(document).on("change", ".document-category", function () {
    var select = $(this);
    $.ajax({
        type: "POST",
        url: "@(Url.Action("Action", "Controller"))",
        data: { id: select.data("id"), 
        tagId: $("option:selected", select).val()},
        error: function () {
            select.after("Error occurred");
        },
        success: function (data) {
            if (data.Success === true) {
                select.after(function () {
                    return $('<div> Saved </div>').delay(1000).fadeOut(1000);
                });
            } else {
                select.after("Error occurred");
            }
        }
    });
});

The very first line can be replaced by $(".document-category").change(function(){ but this time I had to deal with other java-script library that was stripping-bare my stuff. On the other end of the ajax request you need to put controller action:

[HttpPost]
public virtual JsonResult Action(int id, int? tagId)
{
        //blah, do some update stuff
    return Json(new {Success = true, Message = "Updated"});
}