JSFiddle - React, Tailwind, and code Playground

HTML

<script src="http://code.jquery.com/ui/1.11.2/jquery-ui.js"></script>
<link rel="stylesheet" href="http://code.jquery.com/ui/1.11.2/themes/smoothness/jquery-ui.css">
<select id="selectId">
    <option>All options</option>
    <option>Option one</option>
    <option>Option two</option>
</select>

CSS

.ui-selectmenu-button {
    width: auto !important;
}

.ui-state-default .ui-icon.custom-button {
    background-image: url( http://upload.wikimedia.org/wikipedia/commons/5/5f/Triangle-purple.svg);
}

JavaScript

$( "#selectId" ).selectmenu({ icons: { button: "custom-button" } });