Here is a version that works in all modern browsers. The key is using appearance:none which removes the default formatting. Since all of the formatting is gone, you have to add back in the arrow that visually differentiates the select from the input. Note: appearance is not supported in IE.
Working example: https://jsfiddle.net/gs2q1c7p/
select:not([multiple]) {
-webkit-appearance: none;
-moz-appearance: none;
background-position: right 50%;
background-repeat: no-repeat;
background-image: url(data:image/png;base64,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);
padding: .5em;
padding-right: 1.5em
}
#mySelect {
border-radius: 0
}<select id="mySelect">
<option>Option 1</option>
<option>Option 2</option>
</select>Based on Arno Tenkink’s suggestion in the comments, here is an example using a svg instead of a png for the arrow icon.
select:not([multiple]) {
-webkit-appearance: none;
-moz-appearance: none;
background-position: right 50%;
background-repeat: no-repeat;
background-image: url('data:image/svg+xml;utf8,<?xml version="1.0" encoding="utf-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="12" version="1"><path d="M4 8L0 4h8z"/></svg>');
padding: .5em;
padding-right: 1.5em
}
#mySelect {
border-radius: 0
}<select id="mySelect">
<option>Option 1</option>
<option>Option 2</option>
</select>