IE & Firefox – custom drop down could not remove native arrows

Use this it will work but with IE10+ and for FF :

Your css should look like this:

select.desktopDropDown::-ms-expand {
    display: none;
}

More about ::ms-expand.

Then for the rest :

select.desktopDropDown {
    outline : none;
    overflow : hidden;
    text-indent : 0.01px;
    text-overflow : '';
    background : url("../img/assets/arrow.png") no-repeat right #666;

    -webkit-appearance: none;
       -moz-appearance: none;
        -ms-appearance: none;
         -o-appearance: none;
            appearance: none;
}

Note: I hardcoded path "../img/assets/arrow.png" as background.

This should work good for you in IE, Firefox and Opera .