How to get DropDownList SelectedValue in Controller in MVC

1st Approach (via Request or FormCollection):

You can read it from Request using Request.Form , your dropdown name is ddlVendor so pass ddlVendor key in the formCollection to get its value that is posted by form:

string strDDLValue = Request.Form["ddlVendor"].ToString();

or Use FormCollection:

[HttpPost]
public ActionResult ShowAllMobileDetails(MobileViewModel MV,FormCollection form)
{           
  string strDDLValue = form["ddlVendor"].ToString();

  return View(MV);
}

2nd Approach (Via Model):

If you want with Model binding then add a property in Model:

public class MobileViewModel 
{          
    public List<tbInsertMobile> MobileList;
    public SelectList Vendor { get; set; }
    public string SelectedVendor {get;set;}
}

and in View:

@Html.DropDownListFor(m=>m.SelectedVendor , Model.Vendor, "Select Manufacurer")

and in Action:

[HttpPost]
public ActionResult ShowAllMobileDetails(MobileViewModel MV)
{           
   string SelectedValue = MV.SelectedVendor;
   return View(MV);
}

UPDATE:

If you want to post the text of selected item as well, you have to add a hidden field and on drop down selection change set selected item text in the hidden field:

public class MobileViewModel 
{          
    public List<tbInsertMobile> MobileList;
    public SelectList Vendor { get; set; }
    public string SelectVendor {get;set;}
    public string SelectedvendorText { get; set; }
}

use jquery to set hidden field:

<script type="text/javascript">
$(function(){
$("#SelectedVendor").on("change", function {
   $("#SelectedvendorText").val($(this).text());
 });
});
</script>

@Html.DropDownListFor(m=>m.SelectedVendor , Model.Vendor, "Select Manufacurer")
@Html.HiddenFor(m=>m.SelectedvendorText)