I agree with Dave Shenk – if you can specify radius even when border:none, it should be corner-radius rather than border-radius. When specified, the border then just follows the shape of the element.
Definitely specificity by individual corner, and outer/inner border would be very nice. How about border-gradient too? In most instances it’d only be used for 3/4px, but would be a nice smooth fade out and could lead to some much more interesting designs with massive borders and good use of z-index.
Is there any mileage in negative border-radius values? Not so much for “plaque” style corners, but perhaps for a large top-left cut-out into which another rounded box would sit (with a channel between them)…