Remove IE10's "clear field" X button on certain inputs?

Style the ::-ms-clear pseudo-element for the box:

.someinput::-ms-clear {
    display: none;
}

I found it's better to set the width and height to 0px. Otherwise, IE10 ignores the padding defined on the field -- padding-right -- which was intended to keep the text from typing over the 'X' icon that I overlayed on the input field. I'm guessing that IE10 is internally applying the padding-right of the input to the ::--ms-clear pseudo element, and hiding the pseudo element does not restore the padding-right value to the input.

This worked better for me:

.someinput::-ms-clear {
  width : 0;
  height: 0;
}

I would apply this rule to all input fields of type text, so it doesn't need to be duplicated later:

input[type=text]::-ms-clear { display: none; }

One can even get less specific by using just:

::-ms-clear { display: none; }

I have used the later even before adding this answer, but thought that most people would prefer to be more specific than that. Both solutions work fine.